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3. Solving Quadratic Equations
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United States of AmericaTennessee
Integrated Math 2

3.04 The quadratic formula and the discriminant

Lesson
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Lesson

Concept summary

So far the methods we have looked at are useful for quadratic equations that are of a particular form, but we cannot use these methods to solve all quadratic equations. The quadratic formula is a method we can use to solve any quadratic equation written in the form ax^2+bx+c=0 and also to quickly determine the number of real solutions it has.

The quadratic formula is: x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}

The expression under the radical is known as the discriminant.

The square root of a positive number is a real number. The square root of zero is also zero. The square root of a negative number is not a real number. Because of this, we can use the sign of the discriminant to determine the number of real solutions.

b^2-4ac>0

The equation has two real solutions.

Example:

x^2-4x-8=0

b^2-4ac=0

The equation has one real solution.

Example:

9x^2+6x+1=0

b^2-4ac<0

The equation has no real solutions.

Example:

5x^2-3x+9=0

Worked examples

Example 1

Use the discriminant to determine the number and nature of the solutions of the following quadratic equations:

a

2x^2-8x+3=0

Approach

The discriminant is equal to b^2-4ac, and for this equation we have a=2, b=-8, c=3.

Solution

The discriminant is (-8)^2-4(2)(3)=40, which is positive, so the equation has two real solutions.

b

-5x^2+6x-2=0

Approach

The discriminant is equal to b^2-4ac, and for this equation we have a=-5, b=6, c=-2.

Solution

The discriminant has a value of (6)^2-4(-5)(-2)=-4, which is negative, so the equation has no real solutions.

Reflection

As the quadratic formula involves taking the square root of the discriminant, if the discriminant is negative then the formula will involve taking the square root of a negative number, which will in turn result in no real solutions.

We can also see that if 4ac>b^2 the discriminant will be negative, and the corresponding equation will have no real solutions.

c

x^2-6x+9=0

Approach

The discriminant is equal to b^2-4ac, and for this equation we have a=1, b=-6, c=9.

Solution

The discriminant has a value of (-6)^2-4(1)(9)=0, so the equation has one real solution.

Reflection

If the discriminant is zero, the quadratic formula simplifies to be x=-\dfrac{b}{2a}, which is equal to the x-value of the vertex. This means the vertex lies on the x-axis and the x-coordinate of the vertex is the only solution to the quadratic equation.

Example 2

The amount of litter in a park at the end of the day can be modeled against the number of people who visited the park that day by the equation:L=-\frac{1}{50}\left(P^2-73P-150\right) where L is the number of pieces of litter and P is the number of people.

Determine the number of people who visited the park if there are 20 pieces of litter at the end of the day.

Approach

We want to find the number of people, P, when there are 20 pieces of litter at the end of the day, L=20.

We can do this by substituting L=20 into the equation, rearranging the equation into quadratic standard form, and then using the quadratic formula to solve for P.

Solution

\displaystyle L\displaystyle =\displaystyle -\frac{1}{50}\left(P^2-73P-150\right)Model equation
\displaystyle 20\displaystyle =\displaystyle -\frac{1}{50}\left(P^2-73P-150\right)Substitute in L=20
\displaystyle -1000\displaystyle =\displaystyle P^2-73P-150Multiply by -\dfrac{1}{50}
\displaystyle 0\displaystyle =\displaystyle P^2-73P+850Add 1000
\displaystyle P\displaystyle =\displaystyle \frac{73\pm\sqrt{73^2-4(1)(850)}}{2(1)}Quadratic formula

The two solutions (rounded to two decimal places) are P=58.46 and P=14.54. Since we are counting the number of people who visited the park, we want to round to the nearest whole number.

If there are 20 pieces of litter in the park at the end of the day, then either 58 or 15 people visited the park that day.

Reflection

In real life applications where we are counting whole objects, we want to round to the nearest integer so that our solution makes sense in the context.

Outcomes

M2.N.Q.A.1

Use units as a way to understand real-world problems.*

M2.N.Q.A.1.D

Choose an appropriate level of accuracy when reporting quantities.

M2.A.CED.A.1

Create equations and inequalities in one variable and use them to solve problems in a real-world context.*

M2.A.REI.B.2

Solve quadratic equations and inequalities in one variable.

M2.A.REI.B.2.A

Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, knowing and applying the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when a quadratic equation has solutions that are not real numbers.

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP3

Construct viable arguments and critique the reasoning of others.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

M2.MP8

Look for and express regularity in repeated reasoning.

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