Question

show what you know: solving quadratic equations by factori
the solutions of the equation $2x^2 - 5x + 2 = 0$ are \boxed{} and \boxed{}.
the sum of the solutions of the equation $2x^2 - 5x + 2 = 0$ is \boxed{}.
the product of the solutions of the equation $2x^2 - 5x + 2 = 0$ is \boxed{}.

Explanation:

Step1: Factor the quadratic equation

We factor \(2x^{2}-5x + 2 = 0\). We need two numbers that multiply to \(2\times2 = 4\) and add up to \(- 5\). The numbers are \(-1\) and \(-4\). So we rewrite the middle term:
\(2x^{2}-4x - x+2 = 0\)
Group the terms:
\((2x^{2}-4x)-(x - 2)=0\)
Factor out the common factors:
\(2x(x - 2)-1(x - 2)=0\)
Then we get \((2x - 1)(x - 2)=0\)

Step2: Solve for \(x\)

Set each factor equal to zero:

• For \(2x-1 = 0\), we have \(2x=1\), so \(x=\frac{1}{2}\)
• For \(x - 2=0\), we have \(x = 2\)

Step3: Find the sum of the solutions

For a quadratic equation \(ax^{2}+bx + c = 0\), the sum of the roots \(r_1\) and \(r_2\) is given by \(-\frac{b}{a}\). Here \(a = 2\) and \(b=-5\), so the sum is \(-\frac{-5}{2}=\frac{5}{2}\)

Step4: Find the product of the solutions

For a quadratic equation \(ax^{2}+bx + c = 0\), the product of the roots \(r_1\) and \(r_2\) is given by \(\frac{c}{a}\). Here \(a = 2\) and \(c = 2\), so the product is \(\frac{2}{2}=1\)

Answer:

The solutions of the equation \(2x^{2}-5x + 2 = 0\) are \(\frac{1}{2}\) and \(2\).
The sum of the solutions of the equation \(2x^{2}-5x + 2 = 0\) is \(\frac{5}{2}\).
The product of the solutions of the equation \(2x^{2}-5x + 2 = 0\) is \(1\).