Questions: x=(-(-14) ± sqrt((-14)^2-4(1)(-11)))/(2(1))

Transcript text: x=\frac{-(-14) \pm \sqrt{(-14)^{2}-4(1)(-11)}}{2(1)}

Solution

Solution Steps

To solve this problem, we need to identify the correct quadratic formula for the given quadratic equation \(x^2 - 14x - 11 = 0\). The standard quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For the equation \(x^2 - 14x - 11 = 0\), the coefficients are \(a = 1\), \(b = -14\), and \(c = -11\). We need to substitute these values into the quadratic formula and match it with one of the given options.

Step 1: Identify the Coefficients

The given quadratic equation is \(x^2 - 14x - 11 = 0\). From this equation, we identify the coefficients as follows:

\(a = 1\)
\(b = -14\)
\(c = -11\)

Step 2: Calculate the Discriminant

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 \(c\), we get: \[ \Delta = (-14)^2 - 4 \times 1 \times (-11) = 196 + 44 = 240 \]

Step 3: Apply the Quadratic Formula

The quadratic formula to find the roots of the equation is: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substituting the values of \(b\), \(\Delta\), and \(a\), we have: \[ x = \frac{-(-14) \pm \sqrt{240}}{2 \times 1} = \frac{14 \pm \sqrt{240}}{2} \]

Step 4: Calculate the Roots

Calculate the two possible values for \(x\): \[ x_1 = \frac{14 + \sqrt{240}}{2} \approx 14.746 \] \[ x_2 = \frac{14 - \sqrt{240}}{2} \approx -0.746 \]

Final Answer

The roots of the quadratic equation \(x^2 - 14x - 11 = 0\) are: \[ \boxed{x_1 \approx 14.75} \] \[ \boxed{x_2 \approx -0.746} \]

The correct quadratic formula from the given options is: \[ \boxed{x = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(1)(-11)}}{2(1)}} \]

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