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Solve the equation x²=4(x−3)². The diagram shows the sketch of the graph of y = -4x - x² cutting the x-axis at A and B. Find the coordinates of A and the turning point of the graph.

Mathematics

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Answers

2 parts
a

x=2,x=6x=2, x=6

b

A=(−4,0)A=(-4, 0)

(-2, 4)
Graph of y = −4x − x²

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Key points of y = −4x − x²
Pointxy
x-intercept−40
Maximum−24
x-intercept00

Step-by-step solution

10 steps

Part a· 3 steps

Solve the equation x2=4(x−3)2x^2 = 4(x-3)^2.

  1. Take the square root of both sides.

    x2=4(x−3)2\sqrt{x^2} = \sqrt{4(x-3)^2} ∣x∣=2∣x−3∣|x| = 2|x-3|

  2. Consider both positive and negative cases for the absolute values.

    Case 1: x=2(x−3)x = 2(x-3)

    x=2x−6x = 2x - 6 6=2x−x6 = 2x - x x=6x = 6

    Case 2: x=−2(x−3)x = -2(x-3)

    x=−2x+6x = -2x + 6 x+2x=6x + 2x = 6 3x=63x = 6 x=2x = 2

  3. State the solutions.

    The solutions are x=2x=2 and x=6x=6.

    x=2,x=6\boxed{x=2, x=6}

Part b

A=(−4,0)A=(-4, 0)

The diagram shows the sketch of the graph of y=−4x−x2y = -4x - x^2 cutting the x-axis at A and B.

b)(i) Find the coordinates of A.

  1. At the x-axis, the y-coordinate is 0.

    Set y=0y=0 in the equation y=−4x−x2y = -4x - x^2.

    0=−4x−x20 = -4x - x^2

  2. Factor the equation to find the x-intercepts.

    0=x(−4−x)0 = x(-4 - x)

    This gives two possible values for xx:

    x=0or−4−x=0x = 0 \quad \text{or} \quad -4 - x = 0 x=0orx=−4x = 0 \quad \text{or} \quad x = -4

  3. Identify point A from the graph.

    From the diagram, point B is at the origin (0,0)(0,0), and point A is on the negative x-axis. Therefore, A corresponds to x=−4x=-4.
    The coordinates of A are (−4,0)(-4, 0).

    A=(−4,0)\boxed{A=(-4, 0)}

    b)(ii) Find the turning point of the graph.

  4. Identify the coefficients of the quadratic equation y=−x2−4xy = -x^2 - 4x.

    Here, a=−1a = -1 and b=−4b = -4.

  5. Calculate the x-coordinate of the turning point using the formula x=−b2ax = -\frac{b}{2a}.

    x=−−42(−1)x = -\frac{-4}{2(-1)} x=−−4−2x = -\frac{-4}{-2} x=−2x = -2

  6. Substitute the x-coordinate back into the equation y=−4x−x2y = -4x - x^2 to find the y-coordinate.

    y=−4(−2)−(−2)2y = -4(-2) - (-2)^2 y=8−4y = 8 - 4 y=4y = 4

  7. State the coordinates of the turning point.

    The turning point is (−2,4)(-2, 4).

    (−2,4)\boxed{(-2, 4)}
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