Question

Factorise:

10x² - 7ax + (a + 2) (a - 5)
Could anyone solve this for me.
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Answer 1

(5x -a +5)(2x -a -2)

Explanation:

We assume you want to factor this to two linear factors in x and a. A graphing calculator can help immensely (see attached) by showing you the lines you get by setting each factor to zero.

Those lines can be found by considering the x- and a- intercepts.

x-intercepts

Set a=0 and solve for x.

10x² -10 = 0

(x -1)(x +1) = 0 . . . . divide by 10; factor the difference of squares

x = 1 or -1 . . . . . . . . . . zero-product rule

a-intercepts

Set x=0 and solve for a.

(a +2)(a -5) = 0

a = -2 or +5 . . . . . . by the zero-product rule

__

The intercept form of the equation for a line is ...

x/p +y/q = 1 . . . . . . x-intercept = p; y-intercept = q

Possible lines through these intercept points are ...

x/1 +a/-2 = 1 ⇒ 2x -a -2 = 0

x/-1 +a/-2 = 1 ⇒ 2x +a +2 = 0

x/1 +a/5 = 1 ⇒ 5x +a -5 = 0

x/-1 +a/5 = 1 ⇒ 5x -a +5 = 0

These can be paired up two ways. The one of interest can be found by considering the resulting ax term.

(5x +a -5)(2x +a +2) ⇒ ax term is 7ax

(5x -a +5)(2x -a -2) ⇒ ax term is -7ax, as in the given expression

The factorization is (5x -a +5)(2x -a -2).

Answer 2

• (2x - a - 2)(5x - a + 5)

Explanation:

• 10x² - 7ax + (a + 2) (a - 5) =
• 10x² - 7ax + a² - 3a - 10

Factorize by finding the roots

Find the discriminant:

• D = b² - 4ac
• D = (-7a)² - 4*10*(a² - 3a - 10) =
• 49a² - 40a² + 120a + 400 =
• 9a² + 120a + 400 =
• (3a + 20)²

Find the roots:

• x = (- b ± √D)/2a = (7a ± (3a + 20)) / 20

• x₁ = (7a + 3a + 20) / 20 = (10a + 20) / 20 = (a + 2) / 2
• x₂ = (7a - 3a - 20) / 20 = (4a - 20) / 20 = (a - 5) / 5

Factorize the expression:

• 10x² - 7ax + a² - 3a - 10 =
• 10 (x - x₁)(x - x₂) =
• 10[x - (a + 2)/2][[x - (a - 5)/5] =
• (2x - a - 2)(5x - a + 5)

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You would also try a shorter but more difficult way:

• 10x² - 7ax + (a + 2) (a - 5) =
• 2x*5x - 2ax + 10x - 5ax - 10x + (a + 2) (a - 5) =
• 2x*5x - 2x(a - 5) - 5x(a + 2) + (a + 2) (a - 5) =
• 2x(5x - (a - 5)) - (a + 2)(5x - (a - 5)) =
• (5x - a + 5)(2x - a - 2)