GCSE OCR Maths Solving quadratic equations: 22 In this question, all measure

22 In this question, all measurements are in centimetres - OCR - GCSE Maths - Question 22 - 2021 - Paper 3

Question 22

22 In this question, all measurements are in centimetres. The square and the rectangle have the same area. (a) Show that $x^2 - 8x - 20 = 0$. (b) Solve $x^2 - 8x ... show full transcript

Worked Solution & Example Answer:22 In this question, all measurements are in centimetres - OCR - GCSE Maths - Question 22 - 2021 - Paper 3

Step 1

Show that $x^2 - 8x - 20 = 0$.

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Answer

To demonstrate that the equation $x^2 - 8x - 20 = 0$ is valid based on the areas of the square and rectangle, we start by defining the areas of both shapes:

Area of the Square: The area of the square is given by side length squared. Since each side is $x$ , the area is:

$\text{Area of Square} = x^2$
Area of the Rectangle: The area of the rectangle is length times width. The length is given as $2x + 5$ and the width is $4$ . Thus, the area is:

$\text{Area of Rectangle} = (2x + 5) \times 4 = 8x + 20$
Equating Areas: Since the two shapes have the same area, we set their area equations equal to each other:

$x^2 = 8x + 20$
Rearranging the Equation: Rearranging the above equation gives:

$x^2 - 8x - 20 = 0$

Hence, we have shown that the equation $x^2 - 8x - 20 = 0$ holds true.

Step 2

Solve $x^2 - 8x - 20 = 0$.

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Answer

To solve the quadratic equation $x^2 - 8x - 20 = 0$ , we can apply the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where:

$a = 1$ ,
$b = -8$ ,
$c = -20$ .

Calculating the Discriminant:

$b^2 - 4ac = (-8)^2 - 4 \cdot 1 \cdot (-20) = 64 + 80 = 144$
Applying the Quadratic Formula:

$x = \frac{8 \pm \sqrt{144}}{2 \cdot 1} = \frac{8 \pm 12}{2}$
Finding the Solutions:
- First solution:
  $x = \frac{20}{2} = 10$
- Second solution:
  $x = \frac{-4}{2} = -2$

Thus, the solutions to the equation are:

$x = 10$ or $x = -2$ .

22 In this question, all measurements are in centimetres - OCR - GCSE Maths - Question 22 - 2021 - Paper 3

Worked Solution & Example Answer:22 In this question, all measurements are in centimetres - OCR - GCSE Maths - Question 22 - 2021 - Paper 3

Show that $x^2 - 8x - 20 = 0$.

Solve $x^2 - 8x - 20 = 0$.

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