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OCR A-level Maths Quadratic equations

Quadratic equations

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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.

Solving quadratics

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.

Completing the square

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).

The quadratic formula

For any quadratic equation in the form axΒ² + bx + c = 0, the discriminant, bΒ² βˆ’ 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.

The discriminant

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.

Real and repeated roots

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

The factor theorem

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Solve xΒ² + 17x βˆ’ 3 = 2x βˆ’ 39.

OCR A-level Maths Quadratic equations Solve xΒ² + 17x βˆ’ 3 = 2x βˆ’ 39.
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2

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A square of side x βˆ’ 7 has an area of 16. Find x.

OCR A-level Maths Quadratic equations A square of side x βˆ’ 7 has an area of 16. Find x.
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3

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Rearrange y = xΒ² + 22x + 38 into the form y = (x + a)Β² + b.

OCR A-level Maths Quadratic equations Rearrange y = xΒ² + 22x + 38 into the form y = (x + a)Β² + b.
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4

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Rearrange y = 5xΒ² βˆ’ 10x βˆ’ 120 into the form y = a(x + b)Β² + c.

OCR A-level Maths Quadratic equations Rearrange y = 5xΒ² βˆ’ 10x βˆ’ 120 into the form y = a(x + b)Β² + c.
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5

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A square has side 2x βˆ’ 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.

OCR A-level Maths Quadratic equations A square has side 2x βˆ’ 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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6

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Using the quadratic formula, solve 2xΒ² βˆ’ 17x + 3 = 0.

OCR A-level Maths Quadratic equations Using the quadratic formula, solve 2xΒ² βˆ’ 17x + 3 = 0.
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7

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Use the quadratic formula to obtain x for (x βˆ’ 5)/x + (2 + x)/3 = 7.

OCR A-level Maths Quadratic equations Use the quadratic formula to obtain x for (x βˆ’ 5)/x + (2 + x)/3 = 7.
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8

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Given that the area of the given trapezium is 80 unitsΒ², calculate the height of the trapezium using the quadratic formula.

OCR A-level Maths Quadratic equations Given that the area of the given trapezium is 80 unitsΒ², calculate the height of the trapezium using the quadratic formula.
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9

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Find the discriminant of 3xΒ² + 5x βˆ’ 12 = 0 and explain what this says about the number of real roots of the quadratic equation.

OCR A-level Maths Quadratic equations Find the discriminant of 3xΒ² + 5x βˆ’ 12 = 0 and explain what this says about the number of real roots of the quadratic equation.
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10

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Given that the quadratic equation xΒ² βˆ’ 4px + 10 + 7p = 0 has one real root, find the two possible values of p.

OCR A-level Maths Quadratic equations Given that the quadratic equation xΒ² βˆ’ 4px + 10 + 7p = 0 has one real root, find the two possible values of p.
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11

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Find the discriminant of xΒ² βˆ’ 13x βˆ’ 264 = 0 and determine its real roots, if any.

OCR A-level Maths Quadratic equations Find the discriminant of xΒ² βˆ’ 13x βˆ’ 264 = 0 and determine its real roots, if any.
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12

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Determine, using the discriminant, how many real roots the following quadratic equation has: 3xΒ² + 4x + 3 = 2xΒ² βˆ’ x βˆ’ 4. Then, determine the real roots, if any, of the quadratic equation.

OCR A-level Maths Quadratic equations Determine, using the discriminant, how many real roots the following quadratic equation has: 3xΒ² + 4x + 3 = 2xΒ² βˆ’ x βˆ’ 4. Then, determine the real roots, if any, of the quadratic equation.
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13

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Find the discriminant of 64xΒ² βˆ’ 912x + 3249 = 0. Then, determine its real roots, if any and sketch a graph of the quadratic equation.

OCR A-level Maths Quadratic equations Find the discriminant of 64xΒ² βˆ’ 912x + 3249 = 0. Then, determine its real roots, if any and sketch a graph of the quadratic equation.
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14

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Without long division, calculate the remainder when xΒ³ + 7xΒ² + 12x βˆ’ 23 is divided by (x βˆ’ 9).

OCR A-level Maths Quadratic equations Without long division, calculate the remainder when xΒ³ + 7xΒ² + 12x βˆ’ 23 is divided by (x βˆ’ 9).
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15

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Given that (x βˆ’ 1) is a factor of 27xΒ³ + axΒ² βˆ’ 104x βˆ’ 28, find the value of a.

OCR A-level Maths Quadratic equations Given that (x βˆ’ 1) is a factor of 27xΒ³ + axΒ² βˆ’ 104x βˆ’ 28, find the value of a.
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16

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Given that (x βˆ’ 11) is a factor of 15xΒ³ βˆ’ 154xΒ² βˆ’ 119x βˆ’ 22, sketch the graph of y = 15xΒ³ βˆ’ 154xΒ² βˆ’ 119x βˆ’ 22.

OCR A-level Maths Quadratic equations Given that (x βˆ’ 11) is a factor of 15xΒ³ βˆ’ 154xΒ² βˆ’ 119x βˆ’ 22, sketch the graph of y = 15xΒ³ βˆ’ 154xΒ² βˆ’ 119x βˆ’ 22.
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