Quadratic Equations

Standard Form

Meaning: Standard form places all terms on one side.

Definition: A quadratic equation is a$x^{2} + b$$x + c = 0$, where a is not 0.

2$x^{2} + 5$$x - 3 = 0$ is in standard form.

• Move all terms to one side.
• Arrange in descending powers.
• Identify a, b, and c.

Solving by Factorisation

Meaning: Factorisation writes the quadratic as two brackets.

Definition: If ($x - p$)$(x - q) = 0$, then $x = p$ or $x = q$.

$x^{2} - 5$$x + 6 = (x - 2)$($x - 3$), so $x = 2$ or $x = 3$.

• Use zero product rule after factorising.

Quadratic Formula

Meaning: A formula that solves any quadratic equation.

Definition: For a$x^{2} + b$$x + c = 0$, $x = (-b +- \sqrt{b^{2} - 4ac}$) / 2a.

Substitute a, b, and c carefully, including signs.

Discriminant

Meaning: The discriminant tells the nature of roots.

Definition: The discriminant is $D = b^{2} - 4$ac.

If D is positive, there are two real roots. If D is zero, roots are equal. If D is negative, there are no real roots.

Easy

Question: Solve $x^{2} - 5$$x + 6 = 0$.

1. Factorise: $x^{2} - 5$$x + 6 = (x - 2)$($x - 3$).
2. Set each factor to zero.
3. $x = 2$ or $x = 3$.

Medium

Question: Find the discriminant of 2$x^{2} + 3$$x - 5 = 0$.

1. Here $a = 2$, $b = 3$, c = -5.
2. $D = b^{2} - 4$ac.
3. $D = 3^2 - 4$(2)$(-5) = 9 + 40 = 49$.

Exam-level

Question: Solve $x^{2} + 4$$x + 1 = 0$ using the formula.

1. $a = 1$, $b = 4$, $c = 1$.
2. $x = (-4 +- \sqrt{16 - 4}$) / 2.
3. $x = (-4 +- \sqrt{12}$) / 2.
4. x = -2 +- $\sqrt{3}$.