Problem 61 - Algebra 1

Question

The solution of a quadratic equation can be found by graphing each side separately and locating the points of intersection. You may wish to consult page 532 for help in approximating solutions. $$ -2 x^{2}+5 x=8 x^{2}-2 $$

Step-by-Step Solution

Answer

The solutions to the given quadratic equation are the roots obtained from applying the Quadratic Formula, thus the answer is \(x_{1,2} = \frac{-(-5) \pm \sqrt{(-5)^{2}-4*10*2}}{2*10}\).

Step 1: Rearrange the Equation

Rearrange the terms to one side to get a standard quadratic equation. In this case, add \(2x^{2}\) to both sides and add 2 to both sides to obtain: \[10x^{2} - 5x + 2 = 0 \]

Step 2: Apply the Quadratic Formula

The Quadratic Formula is given by:\[x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \]For the given quadratic equation \(10x^{2} - 5x + 2 = 0\), \(a = 10\), \(b = -5\), and \(c = 2\). Substitute these into the Quadratic Formula and simplify to find possible values of x.

Step 3: Calculate Discriminant

The Discriminant \(D\) is \(b^{2}-4ac\). Calculate the value of \(D\). This will determine the nature and number of roots the equation will have. If \(D > 0\), equation will have two distinct roots.

Step 4: Determine the Roots

Use the Discriminant and Quadratic Formula to find the roots. In this case, calculate the value of roots as:\[x_{1,2} = \frac{-(-5) \pm \sqrt{(-5)^{2}-4*10*2}}{2*10}\]

Key Concepts

Quadratic Formula

Solving quadratic equations can initially seem daunting, but the quadratic formula offers a straightforward approach. The formula is \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a \), \( b \), and \( c \) are the coefficients in the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). This formula is comprehensive, meaning it works for any quadratic equation.

To use the quadratic formula, first identify the coefficients \( a \), \( b \), and \( c \) from your quadratic equation. In our scenario, from the equation \( 10x^2 - 5x + 2 = 0 \), we have:

• \( a = 10 \)
• \( b = -5 \)
• \( c = 2 \)

Substitute these values into the formula. This will give you the potential solutions for \( x \). The formula handles both real and complex numbers, providing 'roots' that help us understand where the function crosses the x-axis.

Discriminant

The discriminant is a crucial part of the quadratic formula, contained within the square root, \( \sqrt{b^2 - 4ac} \). It can tell you a lot about the nature of the roots without even solving the equation fully.

To find the discriminant, we calculate \( b^2 - 4ac \). For our example, using the coefficients from the equation \( 10x^2 - 5x + 2 = 0 \), we determine:

• \( b = -5 \)
• \( a = 10 \)
• \( c = 2 \)

Calculating the discriminant gives, \((-5)^2 - 4 \times 10 \times 2 = 25 - 80 = -55 \).

The value of the discriminant helps determine the number and type of roots:

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), the equation has exactly one real root.
• If \( D < 0 \), the equation has two complex roots.

In our example, since \( D = -55 \), we expect two complex roots, indicating the parabola doesn't cross the x-axis.

Roots of Quadratic Equation

The 'roots' of a quadratic equation are the values of \( x \) that satisfy the equation, essentially where the graph intersects the x-axis. Using the quadratic formula, we can find these roots by solving:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For our case, the equation \( 10x^2 - 5x + 2 = 0 \) gave us potential roots through substitution:\[ x_{1,2} = \frac{-(-5) \pm \sqrt{(-5)^{2}-4 \times 10 \times 2}}{2 \times 10} \]This calculates:

• First term simplifies to \( \frac{5 \pm \sqrt{-55}}{20} \).

Given our discriminant was negative, the 'roots' are complex numbers. Complex roots occur in conjugate pairs, meaning they have the form \( p \pm qi \). Here, they do not represent points where the parabola intersects the x-axis, instead, they describe a transformation in the complex number plane.