How to solve a quadratic equation by using the quadratic formula
Suppose you have a quadratic equation of the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0. We can solve this
equation by factoring, completing the square, or by using the quadratic formula.
The quadratic formula states that the solution(s) to the equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 are
given by the equation:
𝒙=
−𝒃±�𝒃𝟐 −𝟒𝒂𝒄
𝟐𝒂
The following problems illustrate how to solve this problem using the quadratic formula.
Example 1: Solve the equation 2𝑥 2 − 3𝑥 = 5
STEP 1: Get the equation in the form 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎 if it’s not already in that form.
The equation 2𝑥 2 − 3𝑥 = 5 is not in this form. We will subtract 5 from both sides and get:
2𝑥 2 − 3𝑥 − 5 = 0
Step 2: If applicable, clear any fractions or decimals from the equation by multiplying by a
common denominator or an appropriate multiple of 10 to get rid of decimals.
2𝑥 2 − 3𝑥 − 5 = 0 does not contain any fractions or decimals so we can skip this step.
STEP 3: Identify what 𝒂, 𝒃, and 𝒄 are.
𝑎 = 2 (coefficient of 𝑥 2 ), 𝑏 = −3( coefficient of x), and 𝑐 = −5 (constant term)
STEP 4: Plug 𝒂, 𝒃, and 𝒄 into the quadratic equation and simplify.
𝑥=
−(−3)±�(−3)2 −4(2)(−5)
2(2)
=
3±√9+40
4
=
3±√49
4
Created by Ivan Canales, Senior Tutor, San Diego City College. [email protected]
Be careful at this step. Many people make arithmetic mistakes at this point specially when
dealing with two minuses.
STEP 5: Simplify further if possible (i.e. simplify square root, divide by a number)
𝑥=
3±√49
4
=
3±7
4
STEP 6: Write down solutions
𝑥=
3+7
4
=
10
4
Hence our solution are 𝑥 = −1,
=
5
2
5
2
or 𝑥
Example 2: Solve the equation
−4
3
=
3−7
4
=
−4
4
= −1
2
𝑥 2 + 2𝑥 − = 0
5
STEP 1: Get the equation in the form 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎 if it’s not already in that form.
The equation
−4
3
2
𝑥 2 + 2𝑥 − = 0 is in this form already so we can skip this step.
5
Step 2: If applicable, clear any fractions or decimals from the equation by multiplying by the
lowest common denominator(lcd) or an appropriate multiple of 10 to get rid of decimals.
The equation
−4
3
2
𝑥 2 + 2𝑥 − = 0 does contain fraction so we need to clear them by
5
multiplying by the lcd. The lcd would be 15.
15●
−4
2
𝑥 2 + 15●2𝑥 − 15● 5 = 15●0
3
Simplifying would give us:
−20𝑥 2 + 30𝑥 − 6 = 0.
STEP 3: Identify what 𝒂, 𝒃, and 𝒄 are.
Created by Ivan Canales, Senior Tutor, San Diego City College. [email protected]
𝑎 = −20 (coefficient of 𝑥 2 ), 𝑏 = 30( coefficient of x), and 𝑐 = −6 (constant term)
STEP 4: Plug 𝒂, 𝒃, and 𝒄 into the quadratic equation and simplify.
𝑥=
−(30) ± �(30)2 − 4(−20)(−6)
−30 ± √900 − 480 −30 ± √420
=
=
2(−20)
−40
−40
STEP 5: Simplify further if possible (i.e. simplify square root, divide by a number)
Note that 420=4●105 so we can simplify the square root because the square root of 4 is 2.
𝑥=
−30 ± √420 −30 ± √4●105 −30 ± 2√105
=
=
−40
−40
−40
Next, notice we can divide everything by 2(i.e. -30, 2√105, -40)
𝑥=
−30±2√105
−40
=
−15±√105
−10
STEP 6: Write down solutions
𝑥=
−15+√105
−10
=
15−√105
10
Hence our solutions are 𝑥 =
or 𝑥 =
−15−√105
−10
15−�105 15+�105
, 10
10
=
15+√105
10
A final note: Your solutions can contain imaginary numbers but all the steps listed are the
same.
If you have any questions about what you have read in this document, please ask a tutor for
further assistance.
Created by Ivan Canales, Senior Tutor, San Diego City College. [email protected]