# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2014 | June | Q#11

**Question**

Given that

f(x) = 2x^{2} + 8x + 3

**a. **find the value of the discriminant of f(x).

**b. **Express f(x) in the form p(x + q)^{2} + r where p, q and r are integers to be found.

The line y = 4x + c, where c is a constant, is a tangent to the curve with equation y = f(x).

**c. **Calculate the value of c.

**Solution**

**a.
**

We are given;

For a quadratic equation , the expression for solution is;

Where is called discriminant.

Therefore;

**b.
**

We have the expression;

We use method of “completing square” to obtain the desired form. We take out factor ‘2’ from the terms which involve ;

Next we complete the square for the terms which involve .

We have the algebraic formula;

For the given case we can compare the given terms with the formula as below;

Therefore we can deduce that;

Hence we can write;

To complete the square we can add and subtract the deduced value of ;

**c. **

If the line y = 4x + c is a tangent to the curve with equation y = f(x), then this line intersects the curve at one and only one point.

If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the curve). Therefore, we can equate coordinates of both lines i.e. equate equations of both the lines (or the line and the curve).

Equation of the line is;

Equation of the curve is;

Equating both equations;

Since line and curve intersect at only one point, solutions of this equation must be a repeated root.

For a quadratic equation , the expression for solution is;

Where is called discriminant.

If , the equation will have two distinct roots.

If , the equation will have two identical/repeated roots.

If , the equation will have no roots.

Therefore, for this equation;

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