Quadratic Formula

Quadratic Formula

Quadratic Formula

Another way to solve quadratic equations.

This works for ALL quadratics (versus factoring that only works with rational solutions)

Quadratic Formula

• To use the formula, you must get the equation in standard form        ( $ax^2+bx+c$  )   

• Just like with factoring, make the  $a$  value positive

•                $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$  

Multiple Choice

Which of the following quadratic equation is written in standard form?

• $5x^2+4-3x=0$

• $12x=7x^2-8$

• $3x^2+5x=-13$

• $3x^2-8x+2=0$

Multiple Choice

What is the value of -(-7)?

• -7

• -8

• 7

• 8

Multiple Choice

What is the value of (-3)^2?

• -6

• -9

• 6

• 9

Fill in the Blank

 $3-5x^2=11x$  

In this quadratic, the  $a$   value is _____?

Fill in the Blank

 $2x^2+x+7=5x^2-3$  

In this quadratic, the  $c$   value is _____?

Fill in the Blank

 $-8x+4x^2-3=4x$  

In this quadratic, the  $b$   value is _____?

Using the Quadratic Formula

Step 1: Set  $=0$   and determine the a, b, and c values

Fill in the Blank

Set the following quadratic $=0$ to use the quadratic formula.

 $-9p^2+6p+11=-4p$  

Identify the  $a$  ,  $b$  , and  $c$  values.  Write them in order with commas between.

Step 2: Substitute the a, b, & c values into the quadratic formula

 $0=9p^2-10p-11$                                                                         
               
    
 $x=\frac{-\left(-10\right)\pm\sqrt{\left(-10\right)^2-4\left(9\right)\left(-11\right)}}{2\left(9\right)}$  

Step 3: Simplify under the radical (simplify the radical once simplified)

 $x=\frac{-\left(-10\right)\pm\sqrt{\left(-10\right)^2-4\left(9\right)\left(-11\right)}}{2\left(9\right)}$  

 $\sqrt{100+396}=\sqrt{496}=\sqrt{4\cdot124}=\sqrt{4\cdot4\cdot31}$  

 $x=\frac{-\left(-10\right)\pm4\sqrt{31}}{2\left(9\right)}$  

Step 5: Simplify the expression fully

 $x=\frac{-\left(-10\right)\pm4\sqrt{31}}{2\left(9\right)}$  

 $x=\frac{10\pm4\sqrt{31}}{18}$  

 $x=\frac{5\pm2\sqrt{31}}{9}$  

Don't forget!!!

 $\pm$   means that there are 2 solutions.  The "+" and the "-"

Fill in the Blank

What are the two solutions for the expression below? 


 $-3\pm5$  

List with a comma between the two solutions

Time to practice!!

Make sure you write the formula down in your notes

so you can complete the next 3 slides.

Multiple Choice

What are the solutions of $a^2=6a+3$ ?

•  $a=\frac{7\pm\sqrt{129}}{2}$  
  

•  $a=3\pm2\sqrt{3}$  
  

•  $a=\frac{7\pm\sqrt{69}}{2}$  
  

•  $a=\frac{-7\pm\sqrt{129}}{2}$  
  

Multiple Choice

What are the solutions of $-4b^2-12b=-6b^2+9-6b$ ?

•  $b=3\pm3\sqrt{2}$  

•  $b=\frac{-5\pm2\sqrt{7}}{3}$  

•  $b=\frac{3\pm3\sqrt{3}}{2}$  

•  $b=\frac{6\pm3\sqrt{11}}{7}$  

Multiple Choice

What are the solutions of $-1=-2x^2+3x$ ?

•  $x=\frac{3\pm\sqrt{17}}{4}$  

•  $x=\frac{-3\pm\sqrt{18}}{4}$  

•  $x=\frac{3\pm\sqrt{17}}{-4}$  

•  $x=\frac{3\pm\sqrt{18}}{2}$  

With practice, it gets easier...

The homework is in Canvas.

Make sure to separate your work for each problem.

These problems should be completed WITHOUT a calculator!