When does $x^2 + ax + a = 1$ have distinct real roots?

When does $x^2 + ax + a = 1$ have distinct real roots?

Question

For what values of the real number $a$ does the quadratic equation $x^2 + ax + a = 1$ have distinct real roots?

1. $a \neq 2$;
2. $a > 2$;
3. $a=2$;
4. all values of $a$.

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[interactive embed (geogebra-iframe): //www.geogebratube.org/material/iframe/id/UCGHBKJ2/width/600/height/400/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/false/ld/false/sri/true/asb/false/at/html5] — not part of the text import; see the source page.

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The red curve here is $y = x^2 + ax + a$, while the blue line is $y = 1$.

What can we say about the curve and the line when the quadratic equation has distinct real roots?

Solution

We can find out about the roots of this quadratic by considering its discriminant.

Before we do this, we must rearrange the equation into the form “$ax^2+bx+c=0$”, so $x^2 + ax + (a-1) = 0.$ Thus the discriminant is equal to $a^2 - 4\times 1 \times (a-1) = a^2 - 4a + 4 = (a-2)^2.$

For the quadratic to have two distinct real roots we need this to be positive and so we need $a \ne 2$.

The answer is (a).