When an SAT question asks how many distinct real solutions a quadratic has, or asks for the value of a constant that gives exactly one solution, it wants the discriminant. You never have to solve the equation. You only compute the piece under the square root in the quadratic formula and check its sign.
The rule
For ax squared plus bx plus c equals zero, the sign of b squared minus 4ac tells you whether there are two, one, or no real solutions.
Write the equation in the form ax squared plus bx plus c equals zero. The discriminant is b squared minus 4ac. If that value is positive, the quadratic has two distinct real solutions. If it is exactly zero, there is one real solution, sometimes called a double root. If it is negative, there are no real solutions, because the formula would ask for the square root of a negative number.
Try x squared plus 6x plus 5 equals zero. Here a is 1, b is 6, and c is 5, so the discriminant is 36 minus 20, which is 16. Positive, so two distinct real solutions. Compare with x squared plus 6x plus 9 equals zero, where the discriminant is 36 minus 36, which is zero, so exactly one solution. And x squared plus 6x plus 11 equals zero gives 36 minus 44, which is negative 8, so no real solutions.
The harder version puts an unknown in the coefficients. Given x squared plus kx plus 25 equals zero has exactly one real solution, set the discriminant to zero: k squared minus 4 times 1 times 25 equals zero, so k squared equals 100 and k is 10 or negative 10. Read the question carefully, because it often asks only for the positive value, and both numbers will be sitting in the choices.
Two setup details cause most of the errors. First, the equation must equal zero before you identify a, b, and c. If you are given 2x squared plus 3x equals 5, move the 5 over to get 2x squared plus 3x minus 5 equals zero, so c is negative 5 rather than 5. Second, b squared is always positive since squaring removes the sign, so a b of negative 7 contributes positive 49.
This idea also answers geometric questions in disguise. A parabola crosses the x-axis twice when the discriminant is positive, touches it once when the discriminant is zero, and never touches it when the discriminant is negative. And for a system made of a line and a parabola, substitute the line into the parabola, collect everything on one side, and the discriminant of that single quadratic tells you how many times the two graphs intersect.