Question

Solve the equation by factoring, by finding square roots, or by using the quadratic formula. $$35 b^{2}-61 b+24=0$$

Short answer

The solution to the equation \(35b^2 - 61b + 24 = 0\) are \(b = 1.14\) and \(b = 0.6\).

Step-by-step solution

Identify the coefficients of the equation

Coefficients of the equation are \(a = 35\), \(b = -61\), and \(c = 24\). Note that these are the constants in the standard form of a quadratic equation \(ax^2 + bx + c = 0\).

Apply the quadratic formula

The quadratic formula, \(x = [-b \pm \sqrt{b^2 - 4ac}]/(2a)\), can be used to find the solutions for the equation. Substituting the values of \(a\), \(b\), and \(c\) into the formula we have \(x = [61 \pm \sqrt{(-61)^2 - (4)(35)(24)}]/(2 * 35)\).

Simplify the Equation

Solving the equation inside the square root yields \(x = [61 \pm \sqrt{(3721) - (3360)}]/(70)\). Further simplifying this gives \(x = [61 \pm \sqrt{361}]/70\) .

Find the roots of the equation

Lastly, solve for \(x\) which will give two solutions \(x = [61 + 19]/70\) and \(x = [61 -19]/70\). Simplifying these gives \(x = 80/70\) and \(x = 42/70\). Therefore, the roots of the quadratic equation (or the values of \(b\)) are \(1.14\) and \(0.6\).