Quadratic Equations in One Variable — chapter notes

A quadratic equation in $x$ is an equation that can be written as $ax^2 + bx + c = 0$, where $a$, $b$, $c$ are real numbers and $a \ne 0$.

• Simplify first: expand brackets and bring every term to one side. If the $x^2$ terms cancel, the equation is not quadratic.
• Read $a$, $b$, $c$ with their signs: $4x^2 = 3 - 7x$ becomes $4x^2 + 7x - 3 = 0$, so $a = 4$, $b = 7$, $c = -3$.
• A root makes the left side $0$; a quadratic equation has at most two roots.

1. Write the equation in standard form.
2. Find two numbers whose product is $ac$ and whose sum is $b$; split $bx$ with them.
3. Group in pairs and take out the common bracket.
4. Put each factor equal to $0$ (the zero-product rule) and solve.

Example: $3x^2 - 10x - 8 = 0$: $ac = -24$, sum $-10$: $-12$ and $2$. $3x(x - 4) + 2(x - 4) = 0$, so $(3x + 2)(x - 4) = 0$ and $x = 4$ or $x = -\dfrac{2}{3}$.

Never divide both sides by $x$ or by a bracket containing $x$: a root is lost.

For $ax^2 + bx + c = 0$ with $b^2 - 4ac \ge 0$, $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$

• Work out $D = b^2 - 4ac$ first.
• $-b$ changes the sign of $b$; $2a$ divides the whole numerator.
• Simplify surds: $\dfrac{6 \pm \sqrt{28}}{4} = \dfrac{6 \pm 2\sqrt{7}}{4} = \dfrac{3 \pm \sqrt{7}}{2}$.

ICSE questions often ask for roots "correct to two decimal places" or "to three significant figures".

• Keep at least four decimal places for the square root (e.g. $\sqrt{33} \approx 5.7446$) and round only the final answer.
• Decimal places count digits after the point: $5.3723 \to 5.37$.
• Significant figures count from the first non-zero digit: $0.7863 \to 0.786$ and $-2.1196 \to -2.12$ (3 s.f.).
• Rounding $\sqrt{29}$ too early ($5.39$) can change the answer: $\dfrac{3 + 5.39}{2} = 4.195$, but the true root $4.1926\ldots$ is $4.19$.

The discriminant of $ax^2 + bx + c = 0$ is $D = b^2 - 4ac$.

• $D > 0$: two real, distinct roots (rational if $D$ is a perfect square, irrational otherwise).
• $D = 0$: two real, equal roots, each $-\dfrac{b}{2a}$; the graph touches the $x$-axis.
• $D < 0$: no real roots; the graph does not meet the $x$-axis.

Example: $2x^2 - 3x + 4 = 0$: $D = 9 - 32 = -23 < 0$, so no real roots.

• Equal roots: put $D = 0$ and solve for the unknown coefficient. Give every value: $3x^2 + kx + 12 = 0$ gives $k^2 = 144$, so $k = \pm 12$.
• Check that the value does not make $a = 0$ (the equation would be linear).
• Real roots: $D \ge 0$; no real roots: $D < 0$. For $x^2 + px + 9 = 0$, no real roots when $-6 < p < 6$.

1. Let $x$ be the unknown and state its units.
2. Form the equation: area = length × breadth; time = distance ÷ speed; work done in one hour = $\dfrac{1}{\text{time alone}}$.
3. Write it in standard form and solve.
4. Reject roots that do not fit (negative lengths, speeds or times; fractional numbers of people; impossible ages) and say why.
5. Answer the question with units, and check in the story.

Example: a $30$ m by $20$ m lawn with an inside path of width $w$ leaves $336$ m²: $(30 - 2w)(20 - 2w) = 336$ gives $w = 3$ or $22$; $22$ is rejected.