To determine which quadratic equations have the solution set {12, 5}, we first need to recognize the properties of quadratic equations and their roots. A quadratic equation can be expressed in the standard form:
ax² + bx + c = 0
According to Vieta’s formulas, if the roots of the quadratic equation are r₁ and r₂, we can express the equation as:
a(x – r₁)(x – r₂) = 0
Substituting the known roots (12 and 5) into this equation gives us:
a(x – 12)(x – 5) = 0
Expanding this, we have:
a(x² – 17x + 60) = 0
Thus, the quadratic equation can take the form:
x² – 17x + 60 = 0
Since any quadratic equation can be multiplied by a non-zero constant ‘a’, the equations:
ax² – 17ax + 60a = 0
for any non-zero value of ‘a’ will also have the same solution set. Therefore, any quadratic equation derived from this condition will be valid.
To identify specific quadratic equations that meet the criteria, check each equation option given in your context. Any equation that simplifies to the form above will have the solution set {12, 5}.