Question

completing the square
in lesson 6.4 we solved quadratic equations using factors and intercept form. but what if cant factor? how can we undo the operations to solve for x?

1. solve each of the following equations.

a. ( x^{2}=64 )
b. ( 4 x^{2}=196 )
c. ( (x + 2)^{2}=25 )

1. a. each of the above equations were solved using a square root. can you square root ( x^{2}+24 x + 144=81 ) in its current form to solve?

b. lets use that to try to rewrite the equation so we can square root. complete the rectangle diagram to find the factors of ( x^{2}+24 x + 144 ) and rewrite the equation with the factors.
c. solve the equation.

1. a. what if the equation doesnt factor to a perfect square? lets try ( 7=x^{2}+6 x ). fill in the rectangle diagram with the ( x^{2} ) and ( x ) terms as if the factors were the same. (the diagram will be incomplete.)

b. what number must go into the last box if the factors were the same? fill it in.
c. if this number were added to the original equation of ( 7=x^{2}+6 x ), what new equation would we have?
d. use the rectangle diagram to write the new equation as a perfect square and solve.

1. lets try another. solve the equation ( 15=x^{2}-14 x )

Explanation:

Step1: Solve \(x^{2}=64\)

Take square root on both sides.
\(x = \pm\sqrt{64}\)

Step2: Solve \(4x^{2}=196\)

First, divide both sides by 4.
\(x^{2}=\frac{196}{4}=49\)
Then take square root.
\(x=\pm\sqrt{49}\)

Step3: Solve \((x + 2)^{2}=25\)

Take square root on both sides.
\(x+2=\pm\sqrt{25}\)
Then solve for \(x\): \(x=-2\pm5\)

Answer:

a. \(x = 8\) or \(x=-8\)
b. \(x = 7\) or \(x=-7\)
c. \(x = 3\) or \(x=-7\)