Discriminant & Solving Quadratics

Discriminant & Solving Quadratics

Unit 4: Factoring & Solving Polynomials

Open Ended

Warm-Up: The height of a ball thrown into the air is represented by the function:

 $h\left(t\right)=-16t^2+30t+6$ 

 Find the roots and determine what they represent. Round to the nearest hundredth. 

Quadratic Equations

Sometimes, we only need to know what TYPE of solutions a quadratic has, not the answers themselves.

Which part of the quadratic formula will determine what type of answers the quadratic has? (Real: rational or irrational; Imaginary)

 $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$  

The DISCRIMINANT

 $b^2-4ac$  is called the discriminant (notice there is NO square root)
The discriminant is a value that can tell us the nature of the roots of a quadratic: 2 real roots (crosses the x-axis twice), 2 imaginary roots (never touches the x-axis), or 1 real root (touches the x-axis once).

The discriminant tells us whether the roots are real or imaginary, and more specifically wheter the REAL roots are rational or irrational. 

Discovering the DISCRIMINANT

I am going to number you off from 1-4. 1s solve #1, 2s solve #2, 3s solve #3, 4s solve #4. Be ready to tell me the discriminant & the roots. 

 $1.\ \ x^2+10x+25=0$  
 $2.\ \ 2x^2-5x-3=0$  
 $3.\ \ 3x^2-11x+2=0$  
 $4.\ \ x^{2\ }-18x+82=0$  

Discovering the DISCRIMINANT

 $1.\ \ x^2+10x+25=0$  
Discriminant: 0       Root: -5
*If the discriminant is 0, you will have ONE, REAL, RATIONAL root. 

 $2.\ \ 2x^2-5x-3=0$  
Discriminant: 49    Roots:   $-\frac{1}{2},\ 3$  
*If the discriminant is a perfect square, you will have TWO, REAL, RATIONAL roots. 

Discovering the DISCRIMINANT

 $3.\ \ 3x^2-11x+2=0$  
Discriminant: 97    Roots:  $\frac{11\pm\sqrt{97}}{6}\approx\ 3.47,\ 0.19$  
*If the discriminant is not a perfect square, you will have TWO, REAL, IRRATIONAL roots. 

 $4.\ \ x^{2\ }-18x+82=0$  
Discriminant: -4    Roots:  $9\pm i$  
*If the discriminant is negative, you will have TWO, IMAGINARY roots.

Multiple Choice

Determine the number and nature of the roots.

• 1 real, rational root

• 2 real, rational roots

• 2 real, irrational roots

• 2 imaginary roots

Multiple Choice

Determine the number and nature of the roots.

• 1 real, rational root

• 2 real, rational roots

• 2 real, irrational roots

• 2 imaginary roots

Multiple Choice

Determine the number and nature of the roots.

• 1 real, rational root

• 2 real, rational roots

• 2 real, irrational roots

• 2 imaginary roots

Multiple Choice

When the discriminant is _________, the quadratic will have 1, real, rational root.

• zero

• positive perfect square

• positive non-perfect square

• negative

Solving Quadratics

20) Notes

There are four methods for solving quadratics:

*Factoring

*Quadratic Formula

*Completing the Square

*Square Root

Solving by Factoring:

Get the equation equal to zero. Factor the quadratic. Then, set each factor equal to zero and solve.

Multiple Choice

Solve the equation by factoring.

• $x=4,\ \frac{6}{5}$

• $x=4,\ -\frac{6}{5}$

• $x=-4,\ \frac{6}{5}$

• $x=-4,\ -\frac{6}{5}$

Solving by Quadratic Formula:

Get the equation equal to zero. Identify a, b, and c. Use the formula to find the roots. 

 $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$  

Multiple Choice

Solve the equation using the quadratic formula.

• $x=\frac{-3\pm\sqrt{69}}{10}$

• $x=\frac{6\pm2\sqrt{69}}{-20}$

• $x=\frac{-3\pm\sqrt{51}}{10}$

• $x=\frac{6\pm2\sqrt{51}}{-20}$

Solving by Completing the Square:

1. Isolate c on the right side of the equation. 

2. Complete the square by adding  $\left(\frac{b}{2}\right)^{2\ }$  to both sides. 
3. Factor the left, simplify the right. 

4. Solve for x. 

Multiple Choice

What is the first step in solving this equation by completing the square.

• Subtract 84 from both sides

• Add 84 to both sides

• Add 8 to both sides

• Complete the square

Multiple Choice

What will we add to both sides to "complete the square"?

• 2

• 4

• -2

• -4

Multiple Choice

In the third step, what does the left side factor into?

• $\left(x+4\right)^2$

• $\left(x-4\right)^2$

• $\left(x+2\right)^2$
  

• $\left(x-2\right)^2$
  

Multiple Choice

Find the solutions. Simplify!

•  $x=-2\pm\sqrt{-88}$  

•  $x=-2\pm2\sqrt{22}$  

•  $x=-1\pm\sqrt{22}$  

•  $x=-2\pm2\sqrt{22}i$  
  

Solving by Taking Square Roots:

Isolate the squared quantity first. Then, square root both sides to begin solving for x. Don't forget $\pm$ !

*This method only works when  $b=0$  

Multiple Choice

Solve the equation using the square root method. Simplify!

•  $\frac{7}{8}$  

•  $\pm\frac{7}{8}$  

•  $\pm\sqrt{\frac{49}{64}}$  

•  $\frac{49}{64}$