A curve has equation y = 4x2 + 13x + 12 A line has equation
y = x + 3
Show that the curve and the line have exactly one point of intersection.
Write down the coordinates of this point of intersection in the form (..., ...)
Do not use a graphical method.
3

(- 1.5, 1.5 )

Step-by-step explanation:

y = 4x² + 13x + 12 → (1)

y = x + 3 → (2)

At the point of intersection the equations are equal, that is

4x² + 13x + 12 = x + 3 ( subtract x + 3 from both sides )

4x² + 12x + 9 = 0 ← in standard form

Consider the factors of the product of the coefficient of the x² term and the constant term which sum to give the coefficient of the x- term.

product = 4 × 9 = 36 and sum = 12

The factors are + 6 and + 6

Use these factors to split the x- term

4x² + 6x + 6x + 9 = 0 ( factor first/second and third/fourth terms )

2x(2x + 3) + 3(2x + 3) = 0 ← factor out (2x + 3) from each term

(2x + 3)(2x + 3) = 0 ← in factored form , that is

(2x + 3)² = 0 , then

2x + 3 = 0 ⇒ 2x = - 3 ⇒ x = - = - 1.5

Substitute x = - 1.5 into (2)

y = - 1.5 + 3 = 1.5

Thus there is only 1 point of intersection at (- 1.5, 1.5 )

what

3

step-by-step explanation:

because where the point where the 2 lines intersect is (3,-1) the x value is 3