To find a quadratic equation with given solutions, we can use the fact that if a quadratic equation has roots (or solutions) of r1 and r2, then it can be expressed in the form:
(x – r1)(x – r2) = 0
In this case, our roots are 5 and 7. Plugging these values into the equation gives us:
(x – 5)(x – 7) = 0
Now, we need to expand this equation.
First, we can use the distributive property:
(x – 5)(x – 7) = x^2 – 7x – 5x + 35
Combining the like terms, we get:
x^2 – 12x + 35 = 0
So the quadratic equation that has solutions of 5 and 7 is:
x^2 – 12x + 35 = 0
This quadratic equation indicates that if you were to solve it using the quadratic formula or factoring, you would arrive back at the roots of 5 and 7.