91Ó°ÊÓ

In quadratic equations, the discriminant is a key player that helps us understand the nature of the roots of the equation. For a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated using the formula:

• \(D = b^2 - 4ac\)

The value of the discriminant allows us to determine the type of roots we will encounter:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root, often called a repeated or double root.
• If \(D < 0\), the roots are not real, but complex or imaginary.

For our given equation \(x^2 - 28x + 187 = 0\), with \(a = 1\), \(b = -28\), and \(c = 187\), the discriminant is:\[D = (-28)^2 - 4 \times 1 \times 187 = 784 - 748 = 36\]Since \(36\) is greater than zero, we know this equation has two different real roots.

The quadratic formula is a powerful tool for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). Regardless of the equation, the formula provides a straightforward method to find the roots. The quadratic formula states:

• \(x = \frac{-b \pm \sqrt{D}}{2a}\)

Here, \(b\) is the coefficient of \(x\), \(a\) is the coefficient of \(x^2\), and \(D\) is the discriminant \(b^2 - 4ac\). "\(\pm\)" indicates that there are two solutions: one with plus and one with minus.For example, with our equation \(x^2 - 28x + 187 = 0\), we know \(a = 1\), \(b = -28\), and \(D = 36\). Plugging these values into the quadratic formula gives us:\[x = \frac{-(-28) \pm \sqrt{36}}{2 \times 1}\]\[x = \frac{28 \pm 6}{2}\]This yields two solutions for \(x\):

• \(x_1 = \frac{28 + 6}{2} = 17\)
• \(x_2 = \frac{28 - 6}{2} = 11\)

The real roots of a quadratic equation are the values of \(x\) that satisfy the equation and are real numbers. When the discriminant \(D\) of a quadratic equation is greater than zero, the equation will have two distinct real roots, which means these solutions are separate numbers and not repeated.In the case of the equation \(x^2 - 28x + 187 = 0\), we determined the discriminant is 36. As a perfect square and greater than zero, it confirms that we have two distinct real roots.To recap, using the quadratic formula, we derived the roots:

• \(x_1 = 17\)
• \(x_2 = 11\)

Both 17 and 11 are real numbers and satisfy the original equation. This means if you substitute these values back into the equation, it holds true for both, confirming they are correct solutions.