Questions: Solve the equation. x^2 - 3x = 40 x=[?] Enter the smallest answer first.

Solution Steps

To solve the quadratic equation \(x^2 - 3x = 40\), we first need to rearrange it into the standard form \(ax^2 + bx + c = 0\). Then, we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots. Finally, we will select the smallest root.

The given equation is: \[ x^2 - 3x = 40 \] We rewrite it in the standard quadratic form \(ax^2 + bx + c = 0\): \[ x^2 - 3x - 40 = 0 \]

From the standard form \(x^2 - 3x - 40 = 0\), we identify the coefficients: \[ a = 1, \quad b = -3, \quad c = -40 \]

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by: \[ \Delta = b^2 - 4ac \] Substituting the values of \(a\), \(b\), and \(c\): \[ \Delta = (-3)^2 - 4 \cdot 1 \cdot (-40) = 9 + 160 = 169 \]

The solutions to the quadratic equation are given by: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substituting the values of \(a\), \(b\), and \(\Delta\): \[ x = \frac{-(-3) \pm \sqrt{169}}{2 \cdot 1} = \frac{3 \pm 13}{2} \]

We calculate the two possible values for \(x\): \[ x_1 = \frac{3 + 13}{2} = \frac{16}{2} = 8.0 \] \[ x_2 = \frac{3 - 13}{2} = \frac{-10}{2} = -5.0 \]

The smallest root is: \[ x_2 = -5.0 \]

Final Answer