The diagram shows part of the graph of $y = x^2 - 2x + 3$ (a) By drawing a suitable straight line, use your graph to find estimates for the solutions of $x^2 - 3x - 1 = 0$. (b) $P$ is the point on the graph of $y = x^2 - 2x + 3$ where $x = 2$. Calculate an estimate for the gradient of the graph at the point $P$.

GCSE Edexcel Maths Coordinate Geometry: The diagram shows part of the gr

The diagram shows part of the graph of $y = x^2 - 2x + 3$ (a) By drawing a suitable straight line, use your graph to find estimates for the solutions of $x^2 - 3x - 1 = 0$ - Edexcel - GCSE Maths - Question 20 - 2017 - Paper 2

Question 20

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The diagram shows part of the graph of $y = x^2 - 2x + 3$ (a) By drawing a suitable straight line, use your graph to find estimates for the solutions of $x^2 -... show full transcript

Worked Solution & Example Answer:The diagram shows part of the graph of $y = x^2 - 2x + 3$ (a) By drawing a suitable straight line, use your graph to find estimates for the solutions of $x^2 - 3x - 1 = 0$ - Edexcel - GCSE Maths - Question 20 - 2017 - Paper 2

Step 1

By drawing a suitable straight line, use your graph to find estimates for the solutions of $x^2 - 3x - 1 = 0$

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Answer

To find the solutions for the equation $x^2 - 3x - 1 = 0$ , we need to graph the equation alongside $y = 0$ .

Rearrange the equation:

$y = x^2 - 3x - 1$
Plot this equation on the same graph as $y = x^2 - 2x + 3$ .
The points where the two graphs intersect represent the solutions to the equation.
Upon visual inspection, we find estimates of the x-values where the intersection occurs. We can get approximate solutions in the range:
- $x ext{ is approximately in the range } -0.8 ext{ to } 0.2$
- $x ext{ is approximately } 2.3 ext{ to } 3.4$ .

Step 2

Calculate an estimate for the gradient of the graph at the point $P$

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Answer

To calculate the gradient at the point $P$ , where $x = 2$ , we need to find the derivative of the function:

The function is:

$y = x^2 - 2x + 3$
Find the derivative:

$\frac{dy}{dx} = 2x - 2$
Substitute $x = 2$ into the derivative:

$\frac{dy}{dx} = 2(2) - 2 = 4 - 2 = 2$
Therefore, the estimate for the gradient at point $P$ is in the range:
- $1.6$ to $2.5$ .

The diagram shows part of the graph of $y = x^2 - 2x + 3$ (a) By drawing a suitable straight line, use your graph to find estimates for the solutions of $x^2 - 3x - 1 = 0$ - Edexcel - GCSE Maths - Question 20 - 2017 - Paper 2

Worked Solution & Example Answer:The diagram shows part of the graph of $y = x^2 - 2x + 3$ (a) By drawing a suitable straight line, use your graph to find estimates for the solutions of $x^2 - 3x - 1 = 0$ - Edexcel - GCSE Maths - Question 20 - 2017 - Paper 2

By drawing a suitable straight line, use your graph to find estimates for the solutions of $x^2 - 3x - 1 = 0$

Calculate an estimate for the gradient of the graph at the point $P$

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