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You can solve some quadratics in the form ax? + bx + c= 0 by factorisation. To factorise a quadratic, try to write it in the form (mx + p)(nx+q)=0 A quadratic equation that can be written in the form Pp q (ms-+ p)(nx+ q)=0 has solutions x=—" orx=—— ~ Find the solutions of the quadratic equation 6x? + 17x +7 =0 by factorisation. Split the xterm so that the two 6¢+17x+7=0 coefficients multiply to give ac. 6x24 3x4 14x+7=0 6x 7=42 and3 x 14 =42 Sx{2x+1)+7(2x+ 1)=0 Factorise the first pair of terms, (2x+ 1)(3x+7)=0 then the second pair of terms, Ree een Take out a factor which is | 2 common to both pairs. Factorise the full expression. A Sometimes a quadratic will not factorise easily. In these cases you may need to complete the square. Any quadratic expression can be written in the | Key point] You'll need to find the following way. This is called completing the square. value of q yourself. It by Be = ee will be equal to c~-— av tberend s+) +q eq! re SSS __ When a= | and q=0, the expression is known as a perfect square, For example, x° + 6x +9 =(x+3)* Perfect squares have only one root, so a graph of the quadratic function touches the x-axis only once, at its vertex. ~ By completing the square, find all the solutions of 4—3x°—6x=0 ro Multiply both sides by —1 i Gerad 0 Manipulate the m = yu! Spe+ es mee expression to obtain a 3[(x+ 1)?-1]-4=0 bracket containing x? 3(x+ 1)??-7=0 and the xterm. 7 Zi +1 =S axel or 2 3 rey te] Complete the square and expand. Substitute this into the previous equation, _/ Completing the square is a useful tool for determining the maximum or minimum point of a quadratic function. @ MyMaths Q 2014-2017, 2025, 2026, 2257 
