A-Level Edexcel Maths Pure Exponential & Logarithms: 4x² + 8x + 3 ≡ a(x + b)² + c (a

4x² + 8x + 3 ≡ a(x + b)² + c (a) Find the values of the constants a, b and c - Edexcel - A-Level Maths Pure - Question 2 - 2013 - Paper 3

Question 2

4x² + 8x + 3 ≡ a(x + b)² + c (a) Find the values of the constants a, b and c. (b) On the axes on page 27, sketch the curve with equation y = 4x² + 8x + 3, showing ... show full transcript

Worked Solution & Example Answer:4x² + 8x + 3 ≡ a(x + b)² + c (a) Find the values of the constants a, b and c - Edexcel - A-Level Maths Pure - Question 2 - 2013 - Paper 3

Step 1

Find the values of the constants a, b and c.

96%

114 rated

Answer

To find the constants a, b, and c in the equation given, we first need to observe the quadratic expression:

$4x^2 + 8x + 3$ .

Identify a: The coefficient of $x^2$ is 4, so we have: $a = 4$ .
Complete the square:

The expression $4x^2 + 8x$ can be rewritten using the completing square method:

$4(x^2 + 2x)$ can be rewritten as:

$4((x + 1)^2 - 1) = 4(x + 1)^2 - 4$

Adding 3 gives: $4(x + 1)^2 - 4 + 3 = 4(x + 1)^2 - 1.$

Hence, the expression becomes: $4(x + 1)^2 - 1.$

We can now clearly see that:
- b is 1 (from $(x + 1)$ )
- c is -1.

Thus, the values are:

$a = 4$
$b = 1$
$c = -1$ .

Step 2

On the axes on page 27, sketch the curve with equation y = 4x² + 8x + 3, showing clearly the coordinates of any points where the curve crosses the coordinate axes.

99%

104 rated

Answer

To sketch the curve defined by the equation $y = 4x^2 + 8x + 3$ , follow these steps:

Find the x-intercepts: Set $y = 0$ :

$4x^2 + 8x + 3 = 0.$

Using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = 4$ , $b = 8$ , $c = 3$ :

$x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 4 \cdot 3}}{2 \cdot 4} = \frac{-8 \pm \sqrt{64 - 48}}{8} = \frac{-8 \pm \sqrt{16}}{8} = \frac{-8 \pm 4}{8}.$

This gives two roots:
- $x = -\frac{1}{2}$
- $x = -2$ .
Find the y-intercept: Set $x = 0$ :

$y = 4(0)^2 + 8(0) + 3 = 3.$ So the curve crosses the y-axis at (0, 3).
Sketch the curve: The curve will:
- Open upwards (since $a > 0$ )
- Cross the y-axis at (0, 3)
- Cross the x-axis at points (-2, 0) and (-0.5, 0).

Make sure to label the intercepts on the sketch to clearly show the coordinates.

4x² + 8x + 3 ≡ a(x + b)² + c (a) Find the values of the constants a, b and c - Edexcel - A-Level Maths Pure - Question 2 - 2013 - Paper 3

Worked Solution & Example Answer:4x² + 8x + 3 ≡ a(x + b)² + c (a) Find the values of the constants a, b and c - Edexcel - A-Level Maths Pure - Question 2 - 2013 - Paper 3

Find the values of the constants a, b and c.

On the axes on page 27, sketch the curve with equation y = 4x² + 8x + 3, showing clearly the coordinates of any points where the curve crosses the coordinate axes.

Join the A-Level students using SimpleStudy...