The quadratic curve $y = x^2 + 4x - 21$ can be written as $y = (x - a)(x + b).$

(a)
Find the value of a and the value of b, where a and b are positive integers.

Answer a = ___

b = ___
(b)
Explain why $y$ will not have a value smaller than $-25$ .

Factorisation quadratic Quadratic properties Quadratic equations Number systems Letters represent numbers 中文

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Worked solution

6(a)

1. Find the two required factors of the quadratic expression.

For $x^2+4x-21$ , find two numbers with product $-21$ and sum $4$ .

The numbers are $-3$ and $7$ , since
$(-3)(7)=-21 \quad \text{and} \quad -3+7=4.$

2. Write the quadratic in factorised form and compare with the given form.

So
$x^2+4x-21=(x-3)(x+7).$

Comparing with
$(x-a)(x+b),$
we get
$a=3,\quad b=7.$

Answer

$a=3$

$b=7$

6(b)

1. Rewrite the quadratic in completed-square form and use the fact that a square is non-negative.

$y=x^2+4x-21$
$y=(x+2)^2-25$
Since $(x+2)^2 \geqslant 0$ for all real values of $x$ ,
$y=(x+2)^2-25 \geqslant -25.$
Therefore, $y$ will not have a value smaller than $-25$ .

Answer

$y=(x+2)^2-25$ . Since $(x+2)^2 \geqslant 0$ , $y \geqslant -25$ . Hence $y$ will not have a value smaller than $-25$ .

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The quadratic curve y=x2+4x−21y = x^2 + 4x - 21y=x2+4x−21 can be written as y=(x−a)(x+b).y = (x - a)(x + b).y=(x−a)(x+b).

Worked solution

6(a)

6(b)

More questions on this topic

The quadratic curve $y = x^2 + 4x - 21$ can be written as $y = (x - a)(x + b).$