Solving Quadratics by Square Roots

Solving Quadratics by Square Roots

Introduction Day 1

When SOLVING quadratic equations ( = ), there is more than 1 method

• Factoring (Quarter 3)

• Square Root Method (TODAY)

• Quadratic formula (next week)

• Completing the Square (maybe later)

What have we learned so far?

Simplifying the Radical

Multiple Choice

√16

• 4

• 8

• 16

• 64

Multiple Choice

Simplify

• 9
• √9
• 3√9
• can't simplify

Multiple Choice

Simplify

 $\sqrt{24}$  

•  $4\sqrt{6}$  

•  $2\sqrt{6}$  

•  $6\sqrt{2}$  

•  $6\sqrt{4}$  

Multiple Choice

Simplify:
√45

• 9√5
• 3√5
• 5√9
• 3√15

To solve quadratic equations, we use inverse (opposite) operations

• The inverse of squared (to the power of 2) is square ROOT

•  $x^2\rightarrow\sqrt{x^2}=x$  

Multiple Choice

 $x^2=9$  

What value of x makes the equation true?

• 3

• 9

• 81

• 4.5

Is there any other value that makes the equation true?           $x^2=9$  

• When we solve using the square root, there will be two possible solutions, the positive value AND the negative value.

•  $\sqrt{x^2}=\sqrt{9}\rightarrow\ x\ =\ 3\ and\ x=-3\ or\ \left\{-3,\ 3\right\}$  

• because  $\left(3\right)^2=9\ and\ \left(-3\right)^2=9,\ so\ x=\pm3$  

Multiple Choice

Solve:  $a^2=4$  

•  $a=2$  and  $a=-2$  

•  $a=2$  

• No solution

•  $a=-2$  

Multiple Choice

Solve:  $k^2=16$  

•  $k=-4$  

•  $k=4$  and  $k=-4$  

•  $k=4$  

• No solution

Multiple Choice

Solve for x

x² = 64

• x = 8, -8

• x = 32, -32

• x = 4, -4

• No real solution

Isolate  $x^2$  then take the square root!

Multiple Choice

Solve:  $n^2-5=-4$  

•  $n=1$  and  $n=-1$  

•  $n=\pm3$  

•  $n=1$  

• No solution

Multiple Choice

Solve using square roots.
k² - 6 = 43

• k = √37
• k = 37 or - 37
• k = 7, k = -7
• no solution

Multiple Choice

Solve:  $m^2+7=88$  

•  $m=\pm\sqrt{95}$  

• No solution

•  $m=\pm9$  

•  $m=\pm2\sqrt{22}$  

Simplify the radical if it is not a perfect square. 

(Don't forget about  $\pm$  still!)

Multiple Choice

Solve:  $x^2+8=28$  

• No solution

•  $x=\pm4\sqrt{5}$  

•  $x=\pm6$  

•  $x=\pm2\sqrt{5}$  

Multiple Choice

Solve:  $g^2+1=19$  

•  $g=\pm2\sqrt{5}$  

•  $g=\pm2\sqrt{3}$  

• No solution

•  $g=3\sqrt{2}$  and  $g=-3\sqrt{2}$  

Multiple Choice

Solve:  $4x^2=112$  

• No solution

•  $x=\pm2\sqrt{7}$  

•  $x=\pm7\sqrt{2}$  

•  $x=\pm6\sqrt{3}$  

Multiple Choice

Solve:  $-3x^2=-63$  

•  $x=\pm\sqrt{21}$  

•  $x=\pm2\sqrt{15}$  

•  $x=\pm\sqrt{66}$  

• No solution

Multiple Choice

Solve:  $7x^2=-21$  

•  $x=\pm\sqrt{3}$  

• No solution

•  $x=\pm2\sqrt{7}$  

•  $x=\pm\sqrt{14}$  

Isolate  $x^2$ completely before 

solving using the square root.

Multiple Choice

Solve by taking square roots:
5m² + 1 = 6

• m = 0
• m = 1 or -1
• m = √7
• m = 7/5 m = -7/5

Multiple Choice

Solve using square roots.

4m² - 100 = 0

• m=25, -25

• m=96

• m=5, -5

• m=-96

Multiple Choice

Solve using square roots.
5b² - 4 = 41

• b = 9, b = -9
• b = 3, b = -3
• b = 8, b = -8
• no solution

Multiple Choice

Solve:  $-3x^2+27=0$  

•  $x=\pm2\sqrt{6}$  

•  $x=\pm\sqrt{30}$  

•  $x=\pm3$  

• No solution

Multiple Choice

Solve the equation

4x² - 25 = 0

• 5/2, -5/2

• 5/2, 5/2

• -5/2, -5/2

• -5/2, 5/2