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Analyzing Quadratics

Analyzing Quadratic Equations

Remember the discriminant is the part of the quadratic formula under the radical. It discriminates between the types of solutions and the types of graphs for quadratic equations. We often call this the "nature" of the roots or solutions. Your nature is how you act and the nature of a quadratic equation is how it "acts".

Take Note!

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b2 − 4ac < 0

D < 0

There are 2 imaginary solutions.

There are 0 x intercepts

A coordinate plane with a graph of 2 parabolas or u shaped graphs. One parabola is above the x-axis opening up, so there are not any x-intercepts. The other parabola is below the x-axis opening down, so there are not any x-intercepts.

b2 − 4ac = 0

There is 1 double root.

There is 1 x intercept.

A coordinate plane with a graph of 2 parabolas or u shaped graphs. One parabola is opening up with the vertex on the x-axis. The other parabola is opening down with the vertex on the x-axis.

b2 − 4ac > 0

D > 0

There are two real roots.

There are 2 x intercepts.

A coordinate plane with a graph of 2 parabolas or u shaped graphs. One parabola is above the x-axis opening down, so there are 2 x-intercepts. The other parabola is below the x-axis opening up, so there are 2 x-intercepts.

Discriminant of the Quadratic Formula

The discriminant of the quadratic formula is the expression under the radical.

x equals the fraction with numerator negative b plus or minus the square root of b squared minus 4 a c and denominator 2 a

In this lesson, you will discover what the discriminant can tell you about the solutions to a quadratic function.

Solutions of a Quadratic Equation

There are three solution possibilities for a quadratic equation.

  1. The quadratic will have two real solutions.
  2. The quadratic will have only one real solution.
  3. The quadratic will have two imaginary solutions.

The discriminant of the quadratic formula will tell you which type of solution possibility you will have.

If b2 − 4ac > 0 Another way to think about this is if b2 − 4ac is a positive value. (i.e. any value greater than zero) , you will have two real solutions.

If b2 − 4ac = 0 If the value is equal to zero... , you will have only one real solution.

If b2 − 4ac < 0 Another way to think about this is if b2 − 4ac is a negative value. (i.e. any value less than zero) , you will have two imaginary solutions.

x-intercepts of a Quadratic

Remember that solutions and x-intercepts are the same thing. The solutions to a quadratic equation are the points of intersection with the x-axis. Therefore, you can expand the previous rules to be:

If b2 − 4ac > 0, you will have two real solutions and two x-intercepts.

If b2 − 4ac = 0, you will have only one real solution and one x-intercept.

If b2 − 4ac < 0, you will have no real solutions and no x-intercepts.

Example #1

Give the discriminant and predict the nature of the roots for the quadratic equations below.

Start by finding the values for a, b, and c from the equation.

−4x2 + 3x − 1 = 0

a=__blank

b=__blank

c=__blank

Answer:

a = −4

b =3

c = −1

2x2 −6x − 2 = 0

a = __blank

b = __blank

c = __blank

Answer:

a = 2

b = −6

c = −3

Now, substitute those values into the equation for the discriminant and solve.

−4x2 + 3x − 1 = 0

a = −4

b = 3

c = −1

D =b2 − 4ac

D =(__blank)2 − 4(__blank)(__blank)

D = __blank − __blank

D = (__blank)

Answer:

D =b2 − 4ac

D =(3)2 − 4(−4)(−1)

D = 9 − 16

D = −7

2x2 − 6x − 2 = 0

a = 2

b = −6

c = −2

D =b2 − 4ac

D =(__blank)2 − 4(__blank)(__blank)

D = __blank + __blank

D = (__blank)

Answer:

D = b2 − 4ac

D =(−6)2 − 4(2)(−2)

D = 36 + 16

D = 52

−4x2 + 3x − 1 = 0

a = −4, b = 3, c = −1

D = −7

Which best describes D? ___blankblankblank___blankblank

How many x-intercepts? ___blankblank

How many roots? ___blankblank

Real or imaginary roots? ___blankblank___blankblank___blankblank_

Answer: D < 0, 0, 2, imaginary

2x2 − 6x − 2 = 0

a = 2, b = −6, c = −2

D = 52

Which best describes D? ___blankblank___blankblank

How many x-intercepts? ___blankblank

How many roots? ___blankblank

Real or imaginary roots? ___blankblank___blankblank___blankblank_

Answer: D > 0, 2, 2, real

Example #2

Determine the value of the discriminant and state the number and type of solutions for 4x2 − 2x + 1 = 0

The formula for the discriminant is b2 − 4ac. What are the values of a, b, and c?

a = ___blankblank, b = ___blankblank, c = ___blankblank

Answer: 4, −2, 1

Evaluate the discriminant for:

a = 4, b = −2, c = 1

b2 − 4ac

( ___blankblank )2 − 4( ___blankblank ) ( ___blankblank )

Answer: (−2)2 − 4(4)(1)

Simplify your expression.

___blankblank − ___blankblank

Answer: 4 − 16

___blankblank

Answer: −12

The value of the discriminant is −12.

How many solutions will the quadratic have?

  • 0
  • 1
  • 2

Answer: 2

How many x-intercepts will the quadratic have?

  • 0
  • 1
  • 2

Answer: 0

Will the solutions be real or imaginary?

  • Real
  • Imaginary

Answer: Imaginary; Because the discriminant is −12, the quadratic will have no (zero) real solutions.

Example #3

Determine the value of the discriminant and state the number and type of solutions for x2 + 5x − 3 = 0

The formula for the discriminant is b2 − 4ac. What are the values of a, b, and c?

a = ___blankblank, b = ___blankblank, c = ___blankblank

Answer: 1, 5, −3

Evaluate the discriminant for:

a = 1, b = 5, c = −3

b2 − 4ac

( ___blankblank )2 − 4( ___blankblank ) ( ___blankblank )

Answer: (5)2 − 4(1)(−3)

Simplify your expression.

___blankblank − ___blankblank

Answer: 25 − 12

___blankblank

Answer: 37

The value of the discriminant is 37.

How many solutions will the quadratic have?

  • 0
  • 1
  • 2

Answer: 2, Because the value of the discriminant is greater than 0, or b2 − 4ac > 0 you will have two real solutions.

How many x-intercepts will the quadratic equations have?

  • 0
  • 1
  • 2

Answer: 2, Because the value of the discriminant is greater than 0, or b2 − 4ac > 0 you will have two x-intercepts.

Will the solutions are real or imaginary?

  • Real
  • Imaginary

Answer: Imaginary