Solving Quadratic Equations using the 4 Methods (Part 1)

Solving Quadratic Equations using the 4 Methods (Part 1)

4 Methods in Solving Quadratic Equations

• Extracting the Square Root

• Factoring

• Completing the Square

• Quadratic Formula

Math Focus 1: Solving Quadratic Eq. by Extracting the Square Root

Just like in solving equations, if we want to find the value of x, we put all the constants on one side, and all the terms with x on the other side. Since quadratic equations contain the term  $x^2$ , we can find the value of x by extracting its square roots.

Note: If it happens that the value of x is a square root of any negative number, then there are no real solutions. In other words, the quadratic equation being solved has no real solutions.

A. Let's Practice!

Solve the quadratic equations by Extracting the Square Root.

Check your answers by substituting the value of x to the given equation.

Multiple Choice

1. Extracting the Square Root.

 $3𝑥^2−4=8$  

•  $x=-1,\ 1$  

•  $x=-2,2$  

•  $x=-3,3$  

•  $x=-4,4$  

Multiple Choice

2. Extracting the Square Root.

 $2(𝑥+2)^2−8=56$ 

•  $x=-4,\ 8$  

•  $x=-8,\ 4$  

•  $x=2\pm4\sqrt{2}$  

•  $x=-2\pm4\sqrt{2}$  

ANSWER KEY

A Let's Practice Activity 1 and 2

Concept Summary

• The standard form of quadratic equation  $ax^2$ + 𝑏𝑥 + 𝑐 = 0 where b is equal to 0 can take the form  $ax^2$ = 𝑐.

• Simplify  $ax^2$ = 𝑐 by combining like terms, if any, until it is reduced to  $x^2$ = 𝑐.

• Determine its two possible roots,  $x=\sqrt{c}$  𝑎𝑛𝑑  $x=-\sqrt{c}$   where c is a nonnegative real number.

• Simplify the roots if necessary and check for accuracy.

Math Focus 2: Solving Quadratic Equations by Factoring

We apply the principle of zero product to determine the solutions of the given problem and other similar problems.

In the principle of zero product, if

𝑎𝑏 = 0, then 𝑎 = 0 𝑜𝑟 𝑏 = 0.

B. Let's Practice!

Solve the Quadratic Equations by Factoring.

Check your answers by substituting the value of x to the given equation.

Multiple Choice

1.By Factoring

 $𝑥^2−7𝑥=0$  

•  $x=7$  

•  $x=0,\ 7$  

•  $x=-7,\ 0$  

•  $x=0$  

Multiple Choice

2.By Factoring

 $2𝑥^2−5𝑥−3=0$  

•  $x=-3,\ \frac{1}{2}$  

•  $x=2,\ 3$  

•  $x=-3,\ 2$  

•  $x=-\frac{1}{2},\ 3$  

ANSWER KEY

B. Let's Practice 1 and 2

Concept Summary

• Here are some suggested steps to be followed in solving quadratic equations by factoring.

• 1. Write the equation in general form  $𝑎𝑥^2+𝑏𝑥+𝑐=0.$  

• 2. Combine like terms if possible.

• 3. Factor the left-hand side of the given equation.

• Use the principle of zero product.

• Solve each resulting linear equation.

• Check each root by substituting it in the original equation.

Math Focus 3: Solving Quadratic Equation by Using the Quadratic Formula

If a quadratic equation is in the form  $𝑎𝑥^2+𝑏𝑥+𝑐=0$  , you can use values for a, b, and c to find the solution of the equation. That is, you can find those values of x that will make the equation true by using the quadratic formula.

The Quadratic Formula

C. Let's Practice!

Solve the quadratic equations by using the Quadratic Formula.

Multiple Choice

1.by using the Quadratic Formula

 $𝑥^2+10𝑥+9=0$  

•  $x=-1,-9$  

•  $x=1,9$  

•  $x=-6,-3$  

•  $x=1,\ 10$  

Multiple Choice

1.by using the Quadratic Formula

 $3𝑥^2+5𝑥=3$  

•  $x=\frac{5\pm\sqrt{61}}{6}$  

•  $x=\frac{-5\pm\sqrt{61}}{6}$  

•  $x=\frac{6\pm\sqrt{56}}{5}$  

•  $x=\frac{-6\pm\sqrt{56}}{5}$  

ANSWER KEY

C. Let's Practice 1 and 2

Concept Summary

A quadratic equation can also have one solution or no real number solution as given in the previous examples.

The following are the steps in solving quadratic equations by using the quadratic formula.

1. Write the equation in standard form (zero on one side of the equation).

2. List the numerical values of the coefficients a,b and c.

3. Write the quadratic formula.

4. Substitute the numerical values for a, b and c in the quadratic formula.

5. Simplify to get the exact solution.

6. Use a calculator, if necessary.

7. Check the roots by substituting them to the original equation.

Formative Assessment

Solving Quadratic Equations by Extracting the Square Root, Factoring and Using the Quadratic Formula

Multiple Choice

1.by Extracting the Square Root

 $5x^2=30.$  

•  $x=\pm6$  

•  $x=\pm\sqrt{6}$  

•  $x=\pm5$  

•  $x=\pm\sqrt{5}$  

Multiple Choice

2.by Extracting the Square Root

 $x^2-36=45$  

•  $x=\pm9$  

•  $x=\pm2$  

•  $x=\pm3$  

•  $no\ real\ roots$  

Multiple Choice

3.by Extracting the Square Root

 $\left(x+3\right)^2-81=0$  

•  $x=\pm9$  

•  $x=6,-12$  

•  $x=-6,12$  

•  $x=9,\ 6$  

Multiple Choice

4.By Factoring

 $x^2-x=0$  .

•  $x=0,\ 1$  

•  $x=0,-1$  

•  $x=0$  

• no real roots
  

Multiple Choice

5.by Factoring

 $x^2+10x+16=0$  

•  $x=-4,-6$  

•  $x=4,6$  

•  $x=2,8$  

•  $x=-8,-2$  

Multiple Choice

6.by Factoring

 $2x^2+3x=2$  

•  $x=-\frac{1}{2},\ 2$  

•  $x=2,\ -1$  

•  $no\ real\ roots$  

•  $x=\frac{1}{2},\ -2$  

Multiple Choice

7.by Quadratic Formula

 $x^2+2x-4=0$  

•  $x=1\pm\sqrt{5}$  

•  $x=2\pm\sqrt{10}$  

•  $x=-1\pm\sqrt{5}$  

•  $-2\pm\sqrt{10}$  

Multiple Choice

8.by Quadratic Formula

 $x^2+7x=8$  

•  $x=-8,1$  

•  $x=-1.-8$  

•  $x=7,\ 8$  

•  $x=-1,-7$  

Multiple Choice

9.by Quadratic Formula

 $2x^2-10x+35=x^2+10$  

•  $x=\pm5$  

•  $x=-1,\ -10$  

•  $x=5$  

•  $x=1,10$  

Poll

How was the lesson?

• I completely understand it.

• I am still a little confused.

• I don't get the lesson at all. I need help.

Study in Advance

Solving Quadratic Equations by Completing the Square