# StudySquare

# OCR A-level Maths Quadratic equations

This page covers the following topics:

Quadratics can be solved by equating one side of the equation to zero and factorising. For a product to be equal to zero, one or both of the factors must be equal to zero. Therefore, the quadratic equation can be solved by equating each factor to zero and solving for the variable.

Quadratic equations that cannot be factorised can be solved by completing the square. Completing the square results in a square plus a term, which can be easily solved.

Another method to solve quadratics which cannot be factorised is by using the quadratic formula. Simply substitute the coefficients of the quadratic equation into the formula to find the solution(s).

For any quadratic equation in the form ax² + bx + c = 0, the discriminant, b² − 4ac, can determine the number of roots the equation has. When the discriminant is positive, the quadratic equation has two distinct real roots. When the discriminant is 0, the quadratic equation has one real root. When the discriminant is negative, the quadratic equation has no real roots. The roots of the quadratic equation can then be calculated using the quadratic formula.

If it is found that the discriminant of a quadratic equation is greater than or equal to zero, ie. the equation has two distinct or equal real roots, the roots can be found by factorising, the quadratic equation or completing the square.

The Remainder Theorem states that if a polynomial, f(x), is divided by (x − a), the remainder is f(a). The Factor Theorem is a special case of the Remainder Theorem; when f(a) = 0, (x − a) is a factor of the polynomial and a is a solution to the polynomial, and vice versa.

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A square has side 2x − 7 and a rectangle has sides x + 9 and 3x + 1. Given that the areas of the two shapes are equal, find the two possible solutions for x.

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Given that the area of the given trapezium is 80 units², calculate the height of the trapezium using the quadratic formula.

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Find the discriminant of 3x² + 5x − 12 = 0 and explain what this says about the number of real roots of the quadratic equation.

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Given that the quadratic equation x² − 4px + 10 + 7p = 0 has one real root, find the two possible values of p.

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Determine, using the discriminant, how many real roots the following quadratic equation has: 3x² + 4x + 3 = 2x² − x − 4. Then, determine the real roots, if any, of the quadratic equation.

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Find the discriminant of 64x² − 912x + 3249 = 0. Then, determine its real roots, if any and sketch a graph of the quadratic equation.

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Without long division, calculate the remainder when x³ + 7x² + 12x − 23 is divided by (x − 9).

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Given that (x − 11) is a factor of 15x³ − 154x² − 119x − 22, sketch the graph of y = 15x³ − 154x² − 119x − 22.

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