Solving Quadratic Equations: Completing the Square

Solving Quadratic Equations: Completing the Square

Complete the Square

Find the "Magic Number"

Start with the "b" value, divide by 2, then square it

6 / 2 = 3 and 3*3 = 9

Balance the equation by adding the "magic number" to both sides.

Steps to solve a quadratic equation using the process of completing the square.

Solve the equation: $x^2+8x-9=0$  

• Isolate the constant: x^2 + 8x = 9

• Find the "magic number" to form a perfect square: 8 / 2 = 4, 4^2 = 16

• Add 16 to both sides: x^2 + 8x + 16 = 9 + 16

• Write the trinomial as a binomial squared: (x + 4)^2 = 25

• Square root both sides: x + 4 = (pos/neg) 5

• Solve the equation: x = -4 plus/minus 5

• Simplify: -4 + 5 = 1, -4 - 5 = -9

Another example

An example where "b" is ODD.

Example with an irrational solution

Analyzing Solutions

The sum of a rational number and an irrational number is always irrational.

Example where "b" is odd and the solution is irrational.

Multiple Select

 $x^2+12x=-20$  

Solve the equation by completing the square. Check the solution(s). 

• x = -4

• x = -2

• x = 4

• x = 2

• x = -10

Multiple Choice

 $x^2+8x\ =\ 33$  

Solve the equation by completing the square. Which is the solution set of the equation?

• {-11, 3}

• {-3, 11}

• {-4, 4}

• {-7, 7}

Multiple Choice

Complete the work shown. Which describes the solutions of the equation?

• Since 7 and 11 are both rational, the sum and difference are rational.

• Since 14 and 11 are both rational, the sum and difference are rational.

• Since 7 is rational and is irrational, the sum and difference are irrational.

• Since is irrational and 11 is rational, the sum and difference are irrational.

Multiple Choice

 $x^2-10=30x$  

Solve the quadratic equation by completing the square.

• {-220, 250}

• {-250, 220}

•  $15\pm\sqrt{235}$  

•  $-15\pm\sqrt{235}$  

Multiple Choice

 $x^2=8-5x$  

Solve the quadratic equation by completing the square.

•  $\frac{5\pm\sqrt{57}}{2}$  

•  $\frac{-5\pm\sqrt{57}}{2}$  

•  $-\frac{47}{4},\ \frac{67}{4}$  

•  $-\frac{67}{4},\ \frac{47}{4}$