Flashcards · 14 cards

Quadratics & their graphs

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Know it
In $y = ax^2 + bx + c$ , what do $a$ and $c$ tell you about the graph?
Answer
$a$ positive: the curve opens upwards (a lowest point). $a$ negative: it opens downwards (a highest point). $c$ is where it cuts the $y$ -axis: the point $(0, c)$ .
From the lesson: The shape of a quadratic
Rule
If $(x - 4)(x + 3) = 0$ , what can you say about $x$ ?
Answer
One of the brackets must be $0$ : $x - 4 = 0$ or $x + 3 = 0$ , so $x = 4$ or $x = -3$ . This works only when the other side is zero.
A rootWhere the curve meets the x-axis, the value is 0
From the lesson: Solving by factorising
Rule
How do you build the quadratic equation whose roots are $3$ and $-\frac25$ ?
Answer
Turn each root into a bracket that is $0$ there: $x = 3$ gives $(x - 3)$ ; $x = -\frac25$ gives $5x = -2$ , so $(5x + 2)$ . Then multiply out: $(x - 3)(5x + 2) = 0$ , that is $5x^2 - 13x - 6 = 0$ .
A rootWhere the curve meets the x-axis, the value is 0
From the lesson: Solving by factorising
Rule
$\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$ . What are their sum and product?
Answer
$\alpha + \beta = -\dfrac{b}{a}$ and $\alpha\beta = \dfrac{c}{a}$ , without solving the equation.
ax² + bx + c = 0, roots α and β
α + β = −b⁄a αβ = c⁄a
The equation is x² − (sum)x + (product) = 0
Sum and product of the rootsSum −b/a, product c/a
From the lesson: Solving by factorising
Rule
Factorise $a^2 - b^2$ .
Answer
$(a + b)(a - b)$ : the difference of two squares. Look for two perfect squares being subtracted.
Difference of two squaresCut b² from a²; the rest rearranges into (a + b)(a − b)
From the lesson: More factorising
Rule
When is $x^2 + bx + c$ a perfect square?
Answer
When $c = \left(\dfrac{b}{2}\right)^2$ , half the coefficient of $x$ , squared. Then $x^2 + bx + c = \left(x + \dfrac{b}{2}\right)^2$ .
(a + b)²The square splits into a², two ab rectangles and b²
From the lesson: Completing the square
Rule
To solve $x^2 + bx = k$ by completing the square, what do you add to both sides?
Answer
$\left(\dfrac{b}{2}\right)^2$ . The left side becomes $\left(x + \dfrac{b}{2}\right)^2$ ; then take the square root of both sides, remembering $\pm$ .
(a + b)²The square splits into a², two ab rectangles and b²
From the lesson: Completing the square
Rule
Write the formula for the roots of $ax^2 + bx + c = 0$ .
Answer
$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . The $\pm$ gives the two roots, the same distance either side of the middle.
Roots on the graphTwo roots, one repeated root, or none
From the lesson: The formula and the discriminant
Rule
What does $b^2 - 4ac$ tell you?
Answer
How many real roots there are. Positive: two different roots. Zero: one repeated root. Negative: no real roots.
The discriminantb² − 4ac > 0, = 0, < 0
From the lesson: The formula and the discriminant
Know it
A parabola crosses the $x$ -axis at $x = -2$ and $x = 3$ . Where is its line of symmetry?
Answer
Exactly halfway between the roots: $x = \dfrac{-2 + 3}{2} = \dfrac12$ . The turning point is on this line.
From the lesson: The shape of a quadratic
Rule
A straight line meets a curve. How do you find the points where they meet?
Answer
Put the line into the curve to get one quadratic in one letter and solve it. Then find the other letter from the line. Two solutions give two meeting points.
A line and a curveThe two crossing points are the two solutions
From the lesson: The formula and the discriminant
Which method?
WAEC 2014 · Paper 2 · Q10 (a)
Solve: $(x - 2)(x - 3) = 12$ .
Can you set each bracket equal to 12?
Answer
No. Brackets can only be set to zero. Expand and make one side zero first: $x^2 - 5x - 6 = 0$ , then factorise.
From the lesson: Solving by factorising Try the question
Which method?
WAEC 2022 · Paper 1 · Q16
What value of $p$ will make $(x^2 - 4x + p)$ a perfect square?
What rule finds $p$ ?
Answer
For a perfect square, the number term is half the coefficient of $x$ , squared: $p = \left(\dfrac{-4}{2}\right)^2$ .
From the lesson: Completing the square Try the question
Which method?
NECO 2024 · Paper 1 · Q33
The difference between the present ages of two brothers is 6 and their product is 135. What is the sum of their ages?
How do you turn this into an equation?
Answer
Two unknowns with a known difference: call the ages $x$ and $x + 6$ . Their product gives a quadratic, $x(x + 6) = 135$ . Solve it, then add the two ages.
From the lesson: The formula and the discriminant Try the question