POP

G7.4 The Quadratic Formula

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01 — Explanation

What does the quadratic formula say, and do I need to memorise it?

For any equation in standard form $ax^2+bx+c=0$ (with $a\neq0$ ):

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

It's given on the Edexcel formula sheet, so don't waste time memorising it — the marks are for correctly identifying $a,b,c$ , substituting carefully (especially negatives), and simplifying methodically.

02 — Explanation

How do I get an equation into standard form before using it?

The equation must equal zero before you can read off $a$ , $b$ , $c$ . Rearrange first:

5x^2=6x+3 \;\Rightarrow\; 5x^2-6x-3=0 \;\Rightarrow\; a=5,\,b=-6,\,c=-3

Exam Tip: Examiners deliberately give equations not already in standard form to test this step. Getting it wrong means every value you substitute afterwards is wrong too.

03 — Explanation

How do I apply the formula step by step?

Worked example: Solve $x^2+2x-35=0$ .

Already in standard form: $a=1,\,b=2,\,c=-35$ .
Substitute (brackets around every value, especially negatives): $x=\dfrac{-(2)\pm\sqrt{(2)^2-4(1)(-35)}}{2(1)}$ .
Discriminant first: $4-(-140)=144$ .
$x=\dfrac{-2\pm\sqrt{144}}{2}=\dfrac{-2\pm12}{2}$ .
Two separate calculations: $x=\dfrac{10}{2}=5$ or $x=\dfrac{-14}{2}=-7$ .

Exam Tip: Calculate the discriminant as its own line of working before substituting it back in — trying to do the whole formula in one calculator line is the most common source of sign errors.

04 — Explanation

What does the discriminant tell me about the roots?

$\Delta=b^2-4ac$ is the value under the root, and its sign decides what kind of solutions exist:

$\Delta$	Roots	Graph of $y=ax^2+bx+c$
$>0$	two distinct real roots	crosses the $x$ -axis twice
$=0$	one repeated root	touches the $x$ -axis once (at the vertex)
$<0$	no real roots	never meets the $x$ -axis

05 — Explanation

How do I use the discriminant to find an unknown coefficient?

Set $\Delta$ equal to, greater than, or less than $0$ depending on the wording, then solve for the unknown.

Worked example: $x^2+kx+9=0$ has equal roots. Find $k$ .

"Equal roots" means $\Delta=0$ : $k^2-4(1)(9)=0 \Rightarrow k^2=36 \Rightarrow k=\pm6$ .

Worked example: $x^2+4x+k=0$ has two distinct real solutions. Find the range of $k$ .

$\Delta>0$ : $16-4k>0 \Rightarrow k<4$ .

06 — Explanation

How do I form and solve a quadratic from a context problem, and interpret the answer?

Worked exam-style question: A trapezium has parallel sides $(x+5)$ and $(3x-2)$ cm and height $(2x-3)$ cm. Its area is $133\text{ cm}^2$ . Show that $8x^2-6x-275=0$ , then find $x$ .

Area formula: $133=\tfrac12\big((x+5)+(3x-2)\big)(2x-3)$ .
Simplify inside: $133=\tfrac12(4x+3)(2x-3)$ , so $266=(4x+3)(2x-3)$ .
Expand: $266=8x^2-6x-9$ , giving $8x^2-6x-275=0$ as required.
Solve with the formula: $a=8,\,b=-6,\,c=-275$ , so $\Delta=36-4(8)(-275)=8836$ , $\sqrt{8836}=94$ .
$x=\dfrac{6\pm94}{16}$ , giving $x=6.25$ or $x=-5.5$ .
Interpret: $x=-5.5$ gives a negative height ( $2(-5.5)-3=-14$ ), which is impossible for a real trapezium — reject it. Only $x=6.25$ is valid.

Exam Tip: Writing down your full substitution line, even with later arithmetic slips, secures the method mark. A correct answer with no working shown can score zero if it happens to be wrong — always show the formula with values substituted in.

07 — Common mistakes

Common mistakes to avoid

Forgetting to rearrange to standard form before reading off $a,b,c$ .
Sign slips in $-b$ or $(-4ac)$ — always substitute negative values in brackets.
Giving both roots in a real-world context without checking whether a negative answer actually makes sense.
Dividing only part of the numerator by $2a$ instead of the whole thing.