Question

Solve the equation by factoring. $$7 x^{2}-10 x+3=0 $$

Short answer

The roots of the quadratic equation \(7x^{2}-10x+3=0\) are \(3/7\) and \(1\).

Step-by-step solution

Factoring

To factorize the given equation \(7x^{2}-10x+3=0\), we have to look for two numbers such that, they multiply to \(7*3=21\) and add up to \(-10\). The numbers \(-3\) and \(-7\) satisfy this condition, because \(-3*-7=21\) and \(-3-7=-10\). So, we rewrite the middle term of the equation as the sum of \(-7x\) and \(-3x\). The equation becomes \(7x^{2}-7x-3x+3=0\).

Grouping and Factoring

Group the first two terms together and the last two terms together, we get \(7x(x-1)-3(x-1)=0\). Now, take out the common factor \((x-1)\), we get \((7x-3)(x-1)=0\)

Solve for x

A product of factors equals zero if at least one of the factors equals zero. Setting each factor equal to zero gives the solutions to the equation. Set \(7x-3 = 0\) and \(x-1=0\). Solving these two equations will give the solutions for the x value.

Find the root of \(7x-3 = 0\)

Solving for x, we have, \(7x = 3\), so \(x = 3/7\). So, \(3/7\) is one root of the equation.

Find the root of \(x-1 = 0\)

Solving for x, we have, \(x = 1\). So, \(1\) is the second root of the given equation.