Quadratic equations of the form ax^2 + bx + c = 0 can have either one or two solutions, called roots. The solutions are found by factoring the equation or using the quadratic formula. When one root is already known, you can use that to help find the other root algebraically.

Let’s look at the equation ax^2 + 5x = 3. We are told one solution is x=1. That means when we substitute x=1 into the equation, it satisfies the equality:

a(1)2 + 5(1) = 3 a + 5 = 3 a = -2

So the full quadratic equation is -2x^2 + 5x = 3. To find the other root, we can factor the left side into (x-1)(x+2) since the roots of this factored form are x=1 and x= -2.

When we multiply it out, we get:

(x-1)(x+2) = -2×2 + 5x

This matches the original equation. Now we can set each factor equal to 0 to find the roots:

x-1 = 0 → x = 1 x+2 = 0 → x = -2

Therefore, the two solutions to the original quadratic equation -2×2 + 5x = 3 are x=1 and x=-2.

We can confirm this by substituting each root back into the original equation:

Let x=1: -2(1)2 + 5(1) = 3 -2 + 5 = 3 ✅

Let x= -2:

-2(-2)2 + 5(-2) = 3 -2(4) + -10 = 3 ✅

In summary, given a quadratic equation ax^2 + bx + c = 0 with one known solution, we can factor it into (x-r1)(x-r2) where r1 is the known root. By setting each factor equal to 0, we find the unknown second root r2. This allows us to algebraically determine both solutions to a quadratic equation when one solution is already provided.

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