[PSAT] 4-Month Revision

Revision of 4th month

Quadratic Equations, Absolute Value, Advanced System of Equations

Multiple Choice

What is the discriminant of $$2x^2+5x-3=0\ ?$$

• 11

• 25

• 49

• 11

Multiple Choice

For $$3x^2-kx+4=0$$ , if D = 25 , find ( k ).

• 5

• 7

• 3

• Can’t be determined

Formula for finding roots

Multiple Choice

Find the roots of $$x^2+6x+8=0$$

• 2,-4

• -2,-4

• -2,4

• 2,4

Viet theorem

​Vieta's Theorem: For quadratic x² + bx + c = 0 (only when a = 1), if the roots are x₁ and x₂, then:

- Sum of roots: x₁ + x₂ = -b

- Product of roots: x₁ × x₂ = c

Use Vieta when a = 1 to quickly find or check roots without using Discriminant

Example: x² - 5x + 6 = 0 → need two numbers that add to 5 and multiply to 6 → x = 2 and x = 3

Multiple Choice

What is the sum of the roots of x² - 9x + 14 = 0?

• 14

• -9

• 9

• -14

Multiple Choice

Find the answers for this quadratic equation x² - 8x + 15 = 0

• 15,3

• 3,5

• 3,8

• 5,8

Finding the number of solutions

​ Number of solutions depends on Discriminant:

( D > 0 ): Two solutions.

( D = 0 ): One solution.

( D < 0 ): No solutions.

Multiple Choice

How many real solutions does the quadratic equation 2x² + 4x + 5 = 0 have?

• 0

• 1

• 2

• 3

Multiple Choice

For the quadratic equation

mx² - 2x + 1 = 0

find the value of m so that the equation has exactly one real solution.

• 1

• 5

• 2

• 7

Factored form of quadratic equations

​Factored Form of Quadratic Equations

A quadratic can be written as y=a(x - x₁)(x - x₂), where x₁ and x₂ are the solutions (roots).

These are the values of x that make y = 0.

Example:

x² - 5x + 6 = 0 factors to (x - 2)(x - 3) = 0, so the solutions are x = 2 and x = 3.

Multiple Choice

What is the factored form of the quadratic equation x² - 7x + 10 = 0?

• (x - 5)(x - 2)

• (x + 5)(x + 2)

• (x - 10)(x + 1)

• (x + 10)(x - 1)

Multiple Choice

For y = 2x² - 8x + 6, find k in y = k(x - m)(x - n).

• 1

• 2

• 3

• 6

Biquadratic equations

What is a Biquadratic Equation?

Form: ax⁴ + bx² + c = 0 (no x³ or x terms!)

How to Solve: The Substitution Method

Key Idea: Turn the hard equation into an easy one!

1. Substitute: Let t = x²

- This changes ax⁴ + bx² + c = 0 into at² + bt + c = 0

- Now you have a simple quadratic equation in t

2. Solve for t: Use any method (factoring, quadratic formula, etc.)

3. Find x: Since t = x², we get x = ±√t

- Each positive t gives you 2 values of x (+ and -)

- Negative t values give no real solutions

Multiple Choice

Solve the biquadratic equation:

x⁴ - 9x² + 20 = 0

• x = ±2, ±√5
• x = ±1, ±√10
• x = ±3, ±√2
• x = ±4, ±√3

Multiple Choice

Solve the equation:

x⁴ + 6x² + 5 = 0

• -1,-5

• 1,5

• Infinitely many solutions

• No solution

Exponential equations that can be converted into quadratic equations

If you see a square root (like √(expression)) equal to something else, square both sides to remove the root.

Then, rewrite or substitute to get a quadratic equation, solve it, and check your answers carefully because squaring can add wrong solutions.

For example, if √(2x - 1) = x + 1, square both sides:

2x - 1 = (x + 1)²

Now solve the quadratic!

Multiple Choice

Solve for x: √(3x + 4) = x - 2

• 4

• 5

• 6

• 3

Multiple Choice

If √(5x - 9) = x + 1, what is the value of 2x?

• 2

• 3

• 6

• 10

Converting word problem into quadratic equations

Quadratic equations can describe real things like areas or falling objects.

Example: The area of a rectangle is 48 square feet. Its length is (x + 4) feet and width is x feet.

The equation is: x(x + 4) = 48 → x² + 4x - 48 = 0

Multiple Choice

The width of a garden is w meters. The length is 4 meters more than the width.

If the area is 45 square meters, what is the width?

• 3

• 5

• 9

• 15

Multiple Choice

A ball is thrown upward, and its height after t seconds is given by:

h = -16t² + 8t + 10
What does the number 10 represent in this equation?

• The initial speed of the ball

• The starting height of the ball

• The acceleration due to gravity

• The time when the ball reaches the ground

Absolute Value

Solving absolute value equations

For |x| = a:

a > 0 : x = a or x = -a

a = 0 : x = 0

a < 0 : No solution

Example: |x + 5| = 3 → x = -2, -8.

Multiple Choice

Find all possible values of x

|4x - 1| = 9.

• x = 3, x=-5/2

• x = -2, x = 5/2

• x = 2, x = -5/2

• x = -2, -5/2

Multiple Choice

If |2x + 3| = m has no solutions, what is true about m?

• m < 0

• m = 0

• m > 0

• m ≥ 0

Solving absolute value inequalities

KEY RULES:

|x| < a (where a > 0): means x is between -a and a → -a < x < a

|x| > a (where a > 0): means x is outside the range → x < -a or x > a

THINK OF IT: |x| is the distance from zero

- |x| < 5 means "distance from zero is less than 5" → between -5 and 5

- |x| > 3 means "distance from zero is more than 3" → beyond -3 or beyond 3

SOLVING STEPS:

1. Isolate the absolute value: |expression| ≤ number

2. Apply the rule: if ≤ or <, write as compound inequality

3. If ≥ or >, write as two separate inequalities with "or"

4. Solve each part

Multiple Choice

Solve |x - 5| > 7.

• x < -2 or x > 12

• -2 < x < 12

• x < 2 or x > 12

• 2 < x < 12

Multiple Choice

Solve |6.5x + 1| < -1

• No solution
• x = 1
• x > 0
• x < -1

Converting word problems into absolute value equations/inequalities

Absolute value measures "distance from target" in real situations with limits or tolerances.

PHRASE TRANSLATIONS:

- At most → ≤

- At least → ≥

- No more than → ≤

- Within → ≤

- Differs by at most → ≤

- Differs by at least → ≥

Multiple Choice

A safe temperature range is within 3°F of 70°F.

Which equation shows this condition?

• |T + 3| ≤ 70

• |T - 70| ≤ 3

• |70 - T| ≥ 3

• |T - 3| ≥ 70

Multiple Choice

The correct password is 2000. If someone enters a number that’s exactly 75 away from it, it’s flagged.

Which equation shows this?

• |x - 75| = 2000

• |x - 2000| = 75

• |x + 75| = 2000

• |x + 2000| = 75

Solving complex system of equations which include squares and cubes

Key Strategies:

Substitution: Isolate a variable and plug it in.

Elimination: Add or subtract equations to cancel terms.

Pattern Use:

Use identities like x³ + y³ = (x + y)(x² - xy + y²)

Tips:

- Use substitution if isolation is easy

- Try test values like 0, 1, or -1

- Watch for quadratics or factorable forms

Multiple Choice

Solve the system:

x² + y = 10

x + y² = 13

What is the 3x+6y?

• 12
• 18
• 21
• 15

Multiple Choice

$$x+y=4\$$

$$x^3+y^3=64$$

What is the value of xy?

• 4

• 2

• 3

• 5

Revision of 3rd month

System of equations, Exponents, and Radicals

System of equations

Real-life application: Systems of equations model real-world scenarios like budgeting or mixing solutions.

Substitution method: Solve one equation for a variable and substitute it into the other to find the solution.

Addition and subtraction method: Add or subtract equations to eliminate a variable and solve for the other.

Identifying suitable method: Choose substitution when a variable is isolated, or elimination when coefficients align easily.

Simplifying equations: Adjust coefficients or clear denominators to make solving easier.

Word problems: Translate real-world scenarios into equations and solve the system.

Easier solving method: Pick the method that requires less work based on the equation structure.

Multiple Choice

A bakery sells cakes for $15 each and pies for $10 each. On a certain day, it sold 20 items and earned $250 in total.

If the bakery had sold 3 fewer cakes and 5 more pies, it would have earned the same total revenue.

How many cakes and pies were sold that day?

• 12 cakes and 8 pies
• 15 cakes and 5 pies
• 5 cakes and 15 pies
• 10 cakes and 10 pies

Multiple Choice

Find 9y-4x from this system of equation:

y = 2x + 3

3x - y = -1.

• 18
• 55
• 42
• 30

Multiple Choice

Find 4x+2y :
2x + 3y = 12
4x - 3y = 6.

• 12
• 16
• 20
• 10

Multiple Choice

John has a total of 9 coins consisting of dimes and quarters, and their total value is $1.50.

How many of the coins are dimes?

• 3
• 7
• 2
• 5

Multiple Choice

Consider the system of equations:

5x - 2y = 10

10x - 4y = 20

Which method would be the easiest to solve this system?

• Substitution

• Elimination

• Both

• Neither

Multiple Choice

Best method for solving this system of equations:
y = 3x - 2
2x + y = 5 ?

• Substitution

• Elimination

• Graphing

• Matrix

Multiple Choice

What steps should you take to eliminate fractions and simplify the system

( ½x + ⅓y = 1 ) and ( ¼x - ⅙y = 2 ) into equations with integer coefficients?

• Multiply the first equation by 6

• Multiply the second equation by 12

• Both A and B

• None

Exponents

- All exponents rules: Exponents follow rules such as xᵐ × xⁿ = xᵐ⁺ⁿ (multiplication),

xᵐ ÷ xⁿ = xᵐ⁻ⁿ (division), and (xᵐ)ⁿ = xᵐⁿ (power of a power).

- Power 0 meaning: Any non-zero number raised to the power of 0 equals 1,

i.e., x⁰ = 1 for x ≠ 0.

- Negative power meaning: A negative exponent means the reciprocal of the base raised

to the positive exponent, i.e., x⁻ⁿ = 1 / xⁿ.

- Identifying the link between “what’s given” and “what is asked”: Recognize the operation

and apply the correct exponent rule—add for multiplication, subtract for division, etc.

- Converting all bases into the same number and solving with exponents: Convert numbers

to the same base (e.g., write 4 as 2² or 9 as 3²) to simplify using exponent rules.

Multiple Choice

Simplify (x³)² ÷ x⁴

• x³
• x⁵
• x
• x²

Multiple Choice

What is the value of (−5)⁰?

• 0
• 1
• -5
• 1.5

Multiple Choice

Simplify 3⁻²

• 1/9
• 9
• 3
• 1/3

Multiple Choice

If x² × xⁿ = x⁵, what is the value of n?

• 4
• 10
• 5
• 6

Multiple Choice

Simplify 2⁵ ÷ 4²

• 2
• 8
• 4
• 1

- Radicals meaning: A radical represents a root of a number, such as √x for the square root

or ∛x for the cube root.

- Proof of radicals concepts: Includes properties like √(a × b) = √a × √b, which are essential

for simplifying radical expressions.

- Converting exponents and radicals: Radicals and exponents are interchangeable.

For example, √x = x¹ᐟ² and x¹ᐟ³ = ∛x.

- Identifying the link between “what’s given” and “what is asked”: Analyze radical expressions

and apply properties or conversions to find the unknown.

- Identifying the whole part of radicals: Estimate square roots by finding the largest integer n

such that n² ≤ x. For example, √65 ≈ 8 because 8² = 64 < 65.

- Rationalizing denominator: To remove radicals from the denominator, multiply numerator

and denominator by a conjugate or suitable radical to simplify.

- Exponential equations (with cube or square roots): To solve, isolate the radical and raise

both sides to the correct power. Example: √x = 4 ⇒ x = 16.

​Radicals

Multiple Choice

What is the value of √25?

• 6
• 4
• 7
• 5

Multiple Choice

Simplify √12 using √(a × b) = √a × √b.

• 2√3
• √6
• 4√3
• 3√4

Multiple Choice

Express 8²ᐟ³ as a radical.

• $$\sqrt{(}8^2)$$

• $$\sqrt[]{(}8^3)$$

• $$(∛8)^2$$

• $$\left(\sqrt{8})^3\right)$$

Multiple Choice

If √x + √y = 4, what is x + y + 2√(xy)?

• 20
• 12
• 16
• 8

Multiple Choice

Simplify $$\frac{2}{\sqrt[]{3}-1}$$

• $\frac{2}{\sqrt{3}+1}$
• $\sqrt{3}-\frac{2}{1}$
• \sqrt{3}+1
• $\sqrt{3}-1$

Multiple Choice

What is the largest integer n such that n² ≤ 40?

• 5
• 6
• 7
• 8

Multiple Choice

What is the largest integer n such that n² ≤ 40?

• 5
• 6
• 7
• 8