Question

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

Answer 1

By equating the equations y = 4x² + 13x + 12 and y = x + 3 and finding the discriminant, which is zero, we prove that there is only one real solution. Thus, the line and the curve intersect at exactly one point, which is (-1.5, 1.5).

Step-by-step explanation:

To prove that the curve y = 4x² + 13x + 12 and the line y = x + 3 intersect at exactly one point without using a graphical method, we must equate the two equations and find the roots of the resultant quadratic equation. If the quadratic has only one real root, then the line and the curve intersect at only one point.

Equating the two equations:

• 4x² + 13x + 12 = x + 3

Rearranging the terms gives us:

• 4x² + 12x + 9 = 0

Now, we calculate the discriminant (Δ) of this quadratic equation:

• Δ = b² - 4ac
• Δ = (12)² - 4(4)(9)
• Δ = 144 - 144
• Δ = 0

Since the discriminant is zero, there is exactly one real solution to this equation, meaning the line and the curve intersect at a single point.

To find the coordinates of the intersection point, we solve for x:

• x = -b/2a
• x = -12/(2*4)
• x = -12/8
• x = -1.5

Plugging x back into the equation of the line to find y:

• y = x + 3
• y = -1.5 + 3
• y = 1.5

Therefore, the point of intersection is (-1.5, 1.5).

Answer 2

(- 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 = -
`(3)/(2)` = - 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 )