In a quadratic expression the highest power term is a square. E.g. x x 2 2x 5x 2 + x - 3

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1 A. Quadratic expressions B. The difference of two squares In a quadratic expression the highest power term is a square. E.g. x + 3x x 5x + x - 3 If a quadratic expression has no x term and both terms are perfect squares and one is being subtracted from the other then this is called the difference of two squares E.g. x 9 x - 1 9x 5 1x 1 5x - 9 Each term is a square C. Factorising the difference of two squares It is very easy to factorise a difference of two squares, just learn to recognise it! Create double brackets Factorise means put into brackets Square root both terms Put both in each bracket with a '+' in one and a '-' in the other a. x = (x + )(x ) b. x 1 = (x + 1)(x 1) c. 9x 5 = (3x + 5)(3x 5) Occasionally we may need to divide by a common factor first d. x = (x 1) = (x + 1)(x 1) e. 3x = 3(x 1) = 3(x + )(x ) f. x 3 = (x 9) = (x + 3)(x 3) D. Factorising other quadratics a. x + x = x(x + ) b. x - 10 = (x - 5) c. x + 9x = 3x(x + 3) E. Factorising quadratics of the form ax + bx + c When these can be factorised it will always be into double brackets When a = 1 put x into both brackets, otherwise we need a product to get ax The product of the numbers in each bracket will be the value of c The sum of the two x terms gives the value of b When a = 1 (x in both brackets) a. x + 3x + = (x + 1)(x + ) 1 x is the only product b. x - 3x + = (x + 1)(x - ) 1 x or x to get -3 c. x - x - = (x + )(x 3) 1 x or x 3 to get -1 d. x + 7x - = (x - 1)(x + ) 1 x or x to get +7 When a > 1 (we need a product of two x terms to get ax ) e. x + 3x - 5 = (x + 5)(x 1) f. 3x - 5x + = (3x - )(x 1) Page 1 of 5

2 F. Solving quadratic equations of the form y = x A quadratic equation usually has solutions a. x - = 17 x = 5 x = ± 5 x = ± 5 b. x + = 1 x = 17 x = ± 17 c. x = x = 3 x = ± 3 G. Solving quadratic equations of the form x + bx = 0 We usually have to make a quadratic equation = 0 to be able to solve it and we sometimes have to factorise a. x + 3x = 0 x(x + 3) = 0 x = 0 or x = -3 The product of numbers = 0 means at least one of the numbers must be 0 b. p = 5p p 5p = 0 p(p 5) = 0 p = 0 or p = 5 H. Solving quadratic equations of the form x + bx + c = 0 c. t = t 0 = t t 0 = t(t ) t = 0 or t = a. x + 5x + = 0 (x + )(x + 1) = 0 x = - or x = -1 b. x + x - 15 = 0 (x + 5)(x - 3) = 0 x = -5 or x = 3 c. x = x x + x - = 0 (x + )(x - ) = 0 x = - or x = Sometimes we have to rearrange to make the equation = 0 Sometimes we are given the information as a problem when we need to make the equation d. The length of a garden is m longer than it s width and the area of the garden is 5m. Form and solve an equation to find the width of the garden. w The length must be w + I. Solving quadratic equations of the form ax + bx + c = 0 (a>1) w(w + ) = 5 w + w = 5 w + w 5 = 0 (w + 9)(w - 5) = 0 w = -9 or w = 5 w = 5m (a width of -9m is nonsensical) a. x + 7x + 3 = 0 (x + 1)(x + 3) = 0 x + 1 = 0 or x + 3 = 0 x = -1 or x = -3 x = - 1 or x = -3 b. x + 5x = 0 (x 1)(3x + ) = 0 x 1 = 0 or 3x + = 0 x = 1 or 3x = - x = 1 or x = - 3 c. x - 3x = 3 x - 3x 3 = 0 3(x x 1) = 0 3(x + 1)(x - 1) = 0 x + 1 = 0 or x - 1 = 0 x = -1 or x = 1 x = - 1 or x = Page of 5

3 J. Solving quadratic equations using the formula If an equation can t be factorised we can solve it using the quadratic formula for ax + bx + c = 0. -b means reverse the sign on b The two solutions are given by x = -b ± b - ac The discriminant b - ac from the quadratic formula is the discriminant When b - ac > 0 there are solutions to the quadratic equation. When b - ac = 0 there is only 1 solution to the quadratic equation. (referred to as a repeated solution) When b - ac < 0 there are no real solutions to the quadratic equation, as you will be trying to find the square root of a negative number. a. x + 5x + = 0 (a = 1 b = 5 c = ) x = -5 ± 5 ( x 1 x ) x 1 x = -5 ± 5 () x = -5 ± 17 x = x = -0. (dp) x = x = -.5 (dp) Press = before you divide! b. 3x + x - = 0 (a = 3 b = c = -) x = - ± ( x 3 x -) x 3 x = - ± 1 (-) x = - ± 0 x = x = x = 0.39 (dp) x = -1.7 (dp) Page 3 of 5

4 c. x - 7x = 5 x - 7x 5 = 0 (a = b = -7 c = -5) x = 7 ± (-7) ( x x -5) x x = 7 ± 9 (-0) x = 7 ± 9 x = x = 7-9 x =.11 (dp) x = -0.1 (dp) d. x - x = 0 (a = b = -1 c = -) x = 1 ± (-1) ( x x -) x x = 1 ± 1 (-3) x = 1 ± 33 x = x = 1-33 x = 0. (dp) x = (dp) K. Completing the square for ax + bx + c = 0 when a = 1 Completing the square is another method to solve a quadratic equation that can't be factorised. We write the quadratic equation in the form (x + m) + n = 0 where m is half the coefficient of x Ensure you have your equation in the order ax + bx + c = 0 before you start a. x + x - = 0 (x + m) + n = 0 m = 3 (x + 3) = 0 (x + 3) = (x + 3)(x + 3) = x + x + 9 So we need -9 as a correction (x + 3) - 11 = 0 (x + 3) = 11 x + 3 = ± 11 x = or x = x = 0.3 (dp) or x = -.3 (dp) b. (x + 1)(x - 5) = 9 x - 5x + x - 5 = 9 x - x - 5 = 9 x - x - 1 = 0 (x + m) + n = 0 m = - (x - ) = 0 (x - ) = (x - )(x - ) = x - x + So we need - as a correction (x - ) - 1 = 0 (x - ) = 1 x - = ± 1 x = or x = x =. (dp) or x = -. (dp) Page of 5

5 L. Completing the square for ax + bx + c = 0 when a 1 Take out a factor of a from the x and x terms then write the starting bracket. Remember m is half the coefficient x. a. x + 3x - = 0 (x + 3 x) - = 0 b. 5x + 3x - 5 = 0 5(x x) - 5 = 0 (x + m) + n = 0 m = 3 5(x + m) + n = 0 m = 3 10 The starting bracket is (x + 3 ) The starting bracket is 5(x ) (x + 3 ) = (x + 3 )(x + 3 ) = (x + x + 9 ) 1 = x + x + 9 = x + 3x + 9 So we need - 9 as a correction (x + 3 ) = 0 (x + 3 ) 3.15 = 0 (x + 3 ) = 3.15 (x + 3 ) = 1.55 x = ± 1.55 x = x = 0.5 or x = x = - 5(x ) = 5(x )(x ) = 5(x + x + 9 ) = 5x + 30 x = 5x + 3x So we need -0.5 as a correction 5(x ) = 0 5(x ) 5.5 = 0 x = (x ) = 5.5 (x ) = 1.09 x = ± 1.09 x = 0.7 (dp) or x = x = -1.3 (dp) Work with fractions or decimals (whatever is the easiest) or a mixture of both until you get to the final answer Page 5 of 5

Finding the Equation of a Graph. I can give the equation of a curve given just the roots.

Finding the Equation of a Graph. I can give the equation of a curve given just the roots. National 5 W 7th August Finding the Equation of a Parabola Starter Sketch the graph of y = x - 8x + 15. On your sketch clearly identify the roots, axis of symmetry, turning point and y intercept. Today