What does b^2-4ac=0 mean?

One repeated real root; the graph touches the x-axis.

What does b^2-4ac<0 mean?

No real roots; the graph does not meet the x-axis.

Complete the square for ax^2+bx+c.

a\left(x+\frac{b}{2a}\right)^2+c-\frac{b^2}{4a}.

State the quadratic formula.

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

Let y=x^2: y=1,4, so x=\pm1,\pm2.

Quadratic functions and the discriminantEdexcel A-Level Maths: Flashcards

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Question

State the discriminant of $ax^2+bx+c$.

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State the discriminant of $ax^2+bx+c$ .: $b^2-4ac$ .
What does $b^2-4ac>0$ mean?: Two distinct real roots.
What does $b^2-4ac=0$ mean?: One repeated real root; the graph touches the $x$ -axis.
What does $b^2-4ac<0$ mean?: No real roots; the graph does not meet the $x$ -axis.
Condition for real roots?: $b^2-4ac\geqslant0$ .
Complete the square for $ax^2+bx+c$ .: $a\left(x+\frac{b}{2a}\right)^2+c-\frac{b^2}{4a}$ .
Turning point of $y=2(x-3)^2-11$ ?: $(3,-11)$ , a minimum.
Line of symmetry of $y=ax^2+bx+c$ ?: $x=-\frac{b}{2a}$ .
State the quadratic formula.: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ .
Solve $x^4-5x^2+4=0$ .: Let $y=x^2$ : $y=1,4$ , so $x=\pm1,\pm2$ .
What substitution solves $2^{2x}-5\times2^x+4=0$ ?: $y=2^x$ , giving $y^2-5y+4=0$ .
Why reject $2^x=-3$ ?: $2^x>0$ for all real $x$ .
Solution of $(p-6)(p+2)\geqslant0$ ?: $p\leqslant-2$ or $p\geqslant6$ .
When does a parabola have a minimum?: When $a>0$ .

Exam questions on Quadratic functions and the discriminant

Consider the equation $x^2+kx+16=0$ , where $k$ is a constant.
Find the set of values of $k$ for which the equation has two distinct real roots.2 marks
The function $f$ is defined by $f(x)=2x^2-12x+7$ for $x\in\mathbb{R}$ .
Hence solve $f(x)=0$ , giving your answers in exact form.2 marks
Consider the equation $x^2+px+(p+3)=0$ , where $p$ is a constant.
Find the set of values of $p$ for which the equation has real roots.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).