Does the equation: x^2+5x-6 have two real roots? If so what are they?

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Having two real roots is equivalent to having two intersections with the x axis.

The equation is a parabola (a U shape) so, having two real roots is equivalent to having the lowest point of the curve below the x axis.

By differentiation we have that 2x+5 is the rate of change, and setting this equal to zero gives the stationary point x=-5/2.

Using the equation we then have:

y=(-5/2)2+5(-5/2)-6=-12.25

This shows that the minimum point of the curve is (-5/2,-12.25) which is below the x axis.

Thus, there are two real roots and applying the quadratic formula gives the values: -6 and 1.

Where the quadratic formula is:

[-b+sqrt(b2-4ac)] / [2a] and [-b-sqrt(b2-4ac)] / [2a]

Stephen B. GCSE Maths tutor, A Level Maths tutor

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