Here are the steps to solve the quadratic equations using the quadratic formula:
1) Write the equation in standard form: ax^2 + bx + c = 0
2) Identify the coefficients a, b, c
3) Plug the coefficients into the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
4) Simplify the solution
The quadratic formula can be used to find the roots (solutions) of any quadratic equation in standard form. Other solution methods like factoring or completing the square may also work for certain equations.
3. The Quadratic Formula
The roots of the polynomial ax 2 + bx + c and the
solutions of the quadratic equation ax 2 + bx + c = 0
-b ± b2 - 4ac
are x = where a ¹ 0 and b2 - 4ac ³ 0
2a
4. Example 1 Find the roots of a polynomial
Find the roots of x2 – 6x + 3.
SOLUTION
The roots of x2 – 6x + 3 are the values of x for which
x2 – 6x + 3 = 0.
–b +
– b2 – 4ac
x= Quadratic formula
2a
– ( – 6) +
– ( – 6)2 – 4( 1 )( 3 ) Substitute values in the quadratic
x= formula: a = 1, b = – 6, and c = 3.
2( 1 )
5. Example 1 Find the roots of a polynomial
6 + 24
= – Simplify.
2
6 +2 6
= – Simplify radical.
2
2(3 + 6 )
–
= = 3 +
– 6 Divide out factor of 2.
2
ANSWER
The roots of x2 – 6x + 3 are 3 + 6 and 3 – 6.
6. Example 1 Find the roots of a polynomial
CHECK Substitute each root for x. The polynomial
should simplify to 0.
(3 + 6 )2 – 6 (3 + 6) + 3
= 9 + 6 6 + 6 – 18 – 6 6 + 3
= 0
(3 – 6 )2 – 6 (3 – 6) + 3
= 9 – 6 6 + 6 – 18 + 6 6 + 3
= 0
7. Example 2 Multiple Choice Practice
Which is one of the solutions to the equation
2x2 – 7 = x?
1 1
– 57 + 57
4 4
–1 + 57 1+ 57
4 4
SOLUTION
Write original
2x2 – 7 = x equation.
Write in standard
2x2 – x – 7 = 0 form.
8. Example 2 Multiple Choice Practice
–b+
– b2 – 4ac Quadratic formula
x =
2a
Substitute values
– ( – 1) + ( – 1)2 – 4( 2 )(–7) in the quadratic
– formula:
=
2( 2 ) a = 2, b = –1, and
c = –7.
1 + 57
–
= Simplify.
4
ANSWER
1 + 57
One solution is .
4
The correct answer is D.
10. Example 4 Choose a solution method
Tell what method(s) you would use to solve the quadratic
equation. Explain your choice(s).
a. 10x2 – 7 = 0
b. x2 + 4x = 0
c. 5x2 + 9x – 4 = 0
SOLUTION
a. The quadratic equation can be solved using square
roots because the equation can be written in the form x2
d. =
11. Example 4 Choose a solution method
b. The quadratic equation can be solved by factoring
because the expression x2 + can be factored easily.
4x
Also, the equation can be solved by completing the
square because the equation is of the form
where a ax2and b is an=even number.
1 + bx + c 0 =
c. The quadratic equation cannot be factored easily, and
completing the square will result in many fractions. So,
the equation should be solved using the quadratic
formula.
12. Example 3 Use the quadratic formula
FILM PRODUCTION
For the period 1971 – 2001, the number y of films
produced in the world can be modeled by the function
y = 10x2 – 94x + 3900 where x is the number of years
since 1971. In what year were 4200 films produced?
SOLUTION
y = 10x2 – 94x + 3900 Write function.
4200 = 10x2 – 94x + 3900 Substitute 4200 for y.
0 = 10x2 – 94x – 300 Write in standard form.
13. Example 3 Use the quadratic formula
Substitute values in the
– ( – 94) +
– ( – 94)2 – 4 (10)(–300) quadratic formula: a = 10,
x =
2(10) b = –94, and c = –300.
94 + 20,836
– Simplify.
=
20
94 + 20,836
The solutions of the equation are ≈ 12 and
20
94 – 20,836
≈ –3.
20
ANSWER
There were 4200 films produced about 12 years after 1971,
or in 1983.
14. 10.7 Warm-up (Day 1)
Use the quadratic formula to Find the roots
1. x 2 + 4x -1
Use the quadratic formula to solve the equation
2. x 2 - 8x +16 = 0
3. x 2 - 5x = -21
4. 4z 2 = 7z + 2
15. 10.7 Warm-up (Day 2)
Use the quadratic formula to solve the equation
1. n2 +1 = 5n
2. 2z + 4 = 3z 2
16. 10.7 Warm-up (Day 3)
Use the quadratic formula to solve the equation
1. 4x 2 + 6x = 3x 2 - 4x +1
2. 7r 2 - 2r = 6