91Ó°ÊÓ

The quadratic formula is an essential tool for solving quadratic equations, which are equations of the form: \( ax^2 + bx + c = 0 \).This formula provides a method to find the roots of the equation and is given by: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).By plugging in the values of the coefficients \(a\), \(b\), and \(c\) into this formula, you can determine the values of \(x\) that satisfy the equation.- **a** is the coefficient of \(x^2\).- **b** is the coefficient of \(x\).- **c** is the constant term.In this example, our equation is \(10y^2 - 16y + 5 = 0\). Therefore, \(a = 10\), \(b = -16\), and \(c = 5\). Simply substitute these into the quadratic formula, and you'll be set for finding the roots.

The discriminant is part of the quadratic formula that helps determine the nature of the roots of a quadratic equation. It is represented by: \( D = b^2 - 4ac \).The value of the discriminant provides useful information:- **If \(D > 0\)**, the equation has two distinct real solutions.- **If \(D = 0\)**, the equation has exactly one real solution (a repeated root).- **If \(D < 0\)**, the equation has no real solutions (the roots are complex).In our exercise, we calculated the discriminant for \(10y^2 - 16y + 5 = 0\) as follows: \((-16)^2 - 4 \cdot 10 \cdot 5 = 256 - 200 = 56\),which means the discriminant is 56, indicating two distinct real roots.

Real solutions of a quadratic equation are the values of \(x\) that satisfy the equation, resulting in true mathematical statements. The number of real solutions depends on the discriminant:- **When the discriminant is positive**, the equation will present two real solutions as demonstrated in our example, where \(D = 56\).The process is to simplify the quadratic formula using the discriminant to find the solutions: \( y = \frac{-(-16) \pm \sqrt{56}}{2 \cdot 10} \).Once calculated and simplified, these provide the real roots of the equation. The steps include computing the square root of the discriminant and performing arithmetic operations to isolate \(y\). Real solutions give insight into where the quadratic expression crosses the x-axis.

Distinct roots refer to the real and separate solutions of a quadratic equation. When the discriminant \(D > 0\), as we have seen in this exercise where \(D = 56\), the quadratic equation has two distinct roots.This means each root value is different. These roots are found using the complete quadratic formula and are calculated as: \[ y = \frac{16 \pm \sqrt{56}}{20} \].Ultimately simplified to: \[ y = \frac{8 \pm \sqrt{14}}{10} \],which gives two different roots:- \( y_1 = \frac{8 + \sqrt{14}}{10} \)- \( y_2 = \frac{8 - \sqrt{14}}{10} \).These roots are distinct, meaning that they do not equal each other, clearly showcasing that they are separated on the x-axis of a graph.