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Quadratic Equation

See how to solve a quadratic equation.
20 examples and their solutions.

Quadratic Equation: Solving by Square Rooting

Example

x2 = 9
Solution

Example

x2 - 7 = 0
Solution

Example

x2 = -2
Solution

Quadratic Equation: Solving by Factoring

Example

x2 - 3x = 0
Solution

Example

x2 + 5x - 14 = 0
Solution

Example

x2 + 12x + 36 = 0
Solution

Example

x2 = 9
(by factoring)
Solution

Quadratic Equation: Completing the Square

Use the completing the square method
when the x coefficient is even.

Example

x2 + 8x - 5 = 0
Solution

Quadratic Formula

Formula

ax2 + bx + c = 0 (a ≠ 0)

→ x = -b ± b2 - 4ac2a
Proof

Example

x2 + 3x - 2 = 0
Solution

Example

4x2 - x + 5 = 0
Solution

Discriminant

Definition

ax2 + bx + c = 0 (a ≠ 0)

→ x = -b ± b2 - 4ac2a

→ D = b2 - 4ac
The discriminant D,
is the square root part
of the quadratic formula.
D determines the nature of the roots
of the quadratic equation.
So you can find the nature of the roots
without solving the quadratic equation.

Nature of the Roots

D > 0, = ■2: 2, rational roots
≠ ■2: 2, irrational roots
D = 0: 1 real root
D < 0: 0 real roots

Example

x2 + 7x + 10 = 0
Nature of the roots?
Solution

Example

x2 - 4x - 1 = 0
Nature of the roots?
Solution

Example

x2 - 6x + 9 = 0
Nature of the roots?
Solution

Example

x2 + 2x + 5 = 0
Nature of the roots?
Solution

Quadratic Equation: Complex Roots

Formula

ax2 + bx + c = 0 (a ≠ 0)

→ x = -b ± b2 - 4ac2a
If you know what a complex number is,
you can find the complex roots
by using the quadratic formula.

Example

4x2 - x + 5 = 0
Solution

Discriminant

D > 0, = ■2: 2, rational roots
≠ ■2: 2, irrational roots
D = 0: 1 real root
D < 0: 2 complex roots (= 0 real roots)
If you know what a complex number is,
you can describe D < 0 differently.

Example

x2 + 2x + 5 = 0
Nature of the roots?
Solution

Quadratic Equation: Sum and Product of the Roots

Formula: Roots → Quadratic Equation

x = r1, r2
→ x2 - (r1 + r2)x + r1r2 = 0
If the roots of a quadratic equation are
r1 and r2,
then the quadratic equation is
x2 - (r1 + r2)x + r1r2 = 0.

Example

x = 3, 4
→ Quadratic equation?
Solution

Example

x = 2 + i
→ Quadratic equation?
Solution

Formula: Quadratic Equation → Sum and Product of the Roots

ax2 + bx + c = 0 (a ≠ 0)

→ r1 + r2 = -ba
r1r2 = ca
From a quadratic equation,
you can directly find
the sum of the roots (r1 + r2)
and the product of the roots (r1r2).

Example

x2 + 6x + c = 0
x = 2
Other root?
Solution

Example

3x2 + bx - 15 = 0
x = 5
Other root?
Solution