Day 6: Using the Quadratic Formula
Not all quadratic equations of the form ax2 + bx + c = 0 can be solved by factoring.
In these situations, the Quadratic Formula must be used.
If ax2 + bx + c = 0 then x =
Example 1:
Solve using the quadratic formula.
a) x2 – 4x – 8 = 0
b) x² + 2x + 1 = 0
c) 2x2 – 5x – 1 = 0
d) 3x2 + 2x + 6 = 0
e) 4x2 – 11x = x – 9
f) x2 – 2x = -3
g) x2 + 4 = 0
When solving quadratic equations, there are three possible outcomes:
1. two real distinct roots
 the graph has 2 x-intercepts
 there are two roots that can be evaluated on a calculator
 the discriminant, D > 0
2. one real distinct root
 the graph has 1 x-intercept
 there is only one root that can be evaluated on a calculator
 the discriminant, D = 0
3. no real roots
 the graph has no x-intercepts
 there are no roots that can be evaluated on a calculator
 the discriminant, D < 0
Recall: D = b2 – 4ac
Example 2:
(the discriminant)
State which outcome occurs for each of the problems in Example 1.
Example 3:
a) Use the quadratic formula to determine the x-intercepts for the function y = x2 + 4x - 12.
b) Use symmetry to determine the coordinates of the vertex of this quadratic function.
c) Sketch the graph of the function, labelling the x-intercepts.
Day 6 Homework:
1. Solve using the quadratic formula. Give exact answers.
a) 6x2 – 7x – 3 = 0
b) 3x2 + 6x + 1 = 0
c) 2x2 + 6x + 3 = 0
d) 3x2 + 7x + 3 = 0
e) X2 + 6x + 4 = 0
2. Determine the exact values of the x-intercepts of each quadratic function. Then approximate
the roots to the nearest hundredth.
a) Y = 2x2 + 5x + 1
b) F(x) = x2 – 6x + 7
1
c) G(x) = 2 𝑥 2 +3x + 6
d) H(x) =
3 2
𝑥 -5x
4
+5
3. Use the discriminant to determine the number of roots for each quadratic equation.
a) x2 – 3x + 1 = 0
b) 3x2 – 6x + 3 = 0
c) 2x2 – 5x + 7 = 0
d) –x2 + 5.5x + 3.25 = 0
e) 5x2 – 10x + 5 = 0
4. Determine the value(s) of k for which the quadratic equation x2 + kx + 4 = 0 will have each
number of roots.
a) One distinct real root (two real, equal roots)
b) Two distinct read roots
c) No real roots
5. The height of a football can be modelled by the function h(t) = -4.9t2 + 21.8t + 1.5, where t is the
time, in seconds, since the ball was thrown, and h is the height of the ball, in metres, above the
ground. Determine how long the football will be in the air, to the nearest tenth of a second.
3
−1
−3±√6
b) = 𝑥 = 3
3
−5±√17
b) 𝑥 = −3 ± √2
4
−3±√3
2
1. a) x = 2 , x = =
c) 𝑥 =
2. a) 𝑥 =
c) 𝑥 = 3 ± √21
−7±√13
e) 𝑥
6
10±2√10
d) 𝑥 =
3
d) 𝑥 =
= −3 ± √5
3. a) 2 distinct real roots b) 1 distinct real root c) no real roots d) two distinct real roots e) 1
distinct real root
4. a) k = 4, k = -4 b) k > 4 or k < -4 c) -4 < k < 4
5. about 4.5 seconds