Chapter 9

Chapter 9

x = 5 or x = -7

Students who answer incorrectly may not have first expanded the right side and brought all terms to one side of the equation

before factoring. Remember that factoring works to solve quadratic equations because of the Zero Product Property. The equation must be written with all terms on one side and a zero on the other before factoring.

Find the vertex using the formula

then find the two x points by factoring

then graph

x = 5 or x = -7

x=0 and x=4

Students who find an incorrect answer may not have correctly applied the Pythagorean Theorem. Draw a diagram of the triangle, labeling each side with its length. Then apply the Pythagorean Theorem. Remember to square each side length first.

Students should keep the context of the problem in mind. Even though there are two solutions to the quadratic equation, one of them results in negative lengths, so it is not a valid solution in

this context.

Trigonometric ratios

28x^2 - 54x - 4 = 0

Solve by factoring. What is the value of x?

x = 2 in.

Solve

5 Seconds

• In standard form, what quadratic equation can be used to find the value of x? 2x^2 + 10x - 13 = 0

• Use the quadratic formula to solve. What are the base and height of Base: 4.3in. Height: 6.1 in.

Students who give an equation of 4x^2 + 20x - 13, may have forgotten to multiply the base and

height of the triangle by 1/2 . Write the formula for the area of a triangle, then substitute the given values and simplify.

Students who give a value of -39 for the discriminant for the second equation, may not have put the equation in standard form. Remember students that the quadratic formula only applies to equations in standard form.

Students who give a value of -39 for the discriminant for the second equation, may not have put the equation in standard form.

Remember students that the quadratic formula only applies to equations in standard form.

• (x - 1)(x + 6) - 3(x - 2) = 99;

• x^2 + 5x - 6 - 3x + 6 = 99;

• x^2 + 2x - 99 = 0;

• (x + 11)(x - 9) = 0; x = -11 or x = 9; distances cannot be negative, so x = 9.

The area of a trapezoid is given by the formula A =1/2(b1 + b2)h, where b1 and b2 are the lengths of the bases and h is the height. A trapezoid has a height equal to its shorter base and a longer base that is 24 cm longer than the shorter base. The area of the trapezoid is 45 square centimeters. Describe how to find the height of

the trapezoid by completing the square. What is the height of the trapezoid?

• 1/2(x + x + 24)x = 45;

• 1/2(2x + 24)x = 45; x2 + 12x = 45;

• x^2 + 12x + 36 = 45 + 36;

• (x + 6)2 = 81; x + 6 = ±9; x + 6 = 9, x = 3 OR x + 6 = -9, x = -15;

• distance must be positive,

  so x = 3.

The side length of a larger square is five times the side length of a smaller square. If the combined area of the two squares is 650 square inches, what is the side length of the larger square? Show your work.

• x^2 + (5x)^2 = 650;

• 26x^2 - 650 = 0;

• x^2 - 25 = 0;

• (x + 5)(x - 5) = 0; x = -5 or x = 5;

• side length must be positive, so x = 5. The side length of the larger square is 5x, or 25 in.

Write a quadratic equation that has no real solutions. In simplified form your equation must have a linear term and a constant term. Explain how you know that the equation has no real solutions.

• x^2 + 4x + 10 = 0; the value of the discriminant is 42 - 4(1)(10) = 16 - 40 = -24.

• If the value of the discriminant is less than 0, then the equation has no real solutions.

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