Problem 67 - Algebra for College Students

Question

Use the quadratic formula and a calculator to solve each equation. Round answers to three decimal places and check your answers. $$x^{2}+3.2 x-5.7=0$$

Step-by-Step Solution

Answer

x ≈ 1.274 or x ≈ -4.474.

Step 1: Identify the coefficients

First, identify the coefficients in the quadratic equation: The equation is given as \[x^{2} + 3.2x - 5.7 = 0\]So, the coefficients are:\[a = 1\] \[b = 3.2\] \[c = -5.7\]

Step 2: Write the quadratic formula

The quadratic formula to find the roots of the equation \[ax^2 + bx + c = 0\] is:\[x = \frac{{-b \pm \sqrt{{b^{2}-4ac}}}}{2a}\]

Step 3: Substitute the values into the formula

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula:\[x = \frac{{-3.2 \pm \sqrt{{3.2^{2} - 4 \cdot 1 \cdot (-5.7)}}}}{2 \cdot 1}\] \[x = \frac{{-3.2 \pm \sqrt{{10.24 + 22.8}}}}{2}\] \[x = \frac{{-3.2 \pm \sqrt{{33.04}}}}{2}\]

Step 4: Calculate the discriminant

Calculate the value under the square root (the discriminant):\[\sqrt{{33.04}} \approx 5.749\]

Step 5: Calculate the two possible values of x

Now calculate the two possible values for \(x\): \[x_1 = \frac{{-3.2 + 5.749}}{2} \approx 1.274\] \[x_2 = \frac{{-3.2 - 5.749}}{2} \approx -4.474\]

Step 6: Check the solutions

Substitute \(x_1\) and \(x_2\) back into the original equation to verify:For \(x_1 = 1.274\): \[((1.274)^{2} + 3.2(1.274) - 5.7) \approx 0\] For \(x_2 = -4.474\): \[(( -4.474)^{2} + 3.2( -4.474) - 5.7) \approx 0\] Both values approximately satisfy the equation.

Key Concepts

quadratic equations

Quadratic equations are mathematical expressions that involve terms up to the second degree. They have the general form:

• 
  
  \( ax^2 + bx + c = 0 \)

Here, 'a', 'b', and 'c' are constants, and 'x' is the variable. The term \(ax^2\) is the quadratic component, \(bx\) is the linear component, and 'c' is the constant term. These equations are used to model various real-world problems like projectile motion, area optimization, and much more. Solving quadratic equations typically involves finding the values of 'x' that satisfy the equation.

solving equations

To solve quadratic equations, one reliable method is using the quadratic formula. This formula lets you find the solutions (or roots) of the equation, and it works even for complex and irrational numbers. The quadratic formula is:

• 
  \( x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{2a} \)

In the formula, 'a', 'b', and 'c' are the coefficients from the quadratic equation \(ax^2 + bx + c = 0\). The formula includes a '±' symbol, which means there will generally be two solutions: one for the plus and one for the minus. To apply this formula, simply substitute the coefficients into the formula and calculate the values. This will give you the possible values of 'x' that solve the equation.

discriminant

The discriminant is an important part of the quadratic formula and it helps to determine the nature of the roots. It is given by the expression:

• 
  \( b^2 - 4ac \)

The value of the discriminant provides crucial information:

• If \( b^2 - 4ac > 0 \) : There are two distinct real roots.
• If \( b^2 - 4ac = 0 \) : There is exactly one real root (a repeated root).
• If \( b^2 - 4ac < 0 \) : There are no real roots, but two complex roots.

In the example \(x^2 + 3.2x - 5.7 = 0\), the discriminant is calculated as \(3.2^2 - 4 \times 1 \times -5.7 = 33.04\), which is greater than 0, indicating two distinct real roots.

roots of equations

The roots of a quadratic equation are the solutions for 'x' that make the equation equal to zero. Using the quadratic formula, we find these roots by solving:

• \( x_1 = \frac{{-b + \sqrt{b^2 - 4ac}}}{2a} \)
• \( x_2 = \frac{{-b - \sqrt{b^2 - 4ac}}}{2a} \)

In our example equation, \(x^2 + 3.2x - 5.7 = 0\), the roots are found to be:

• \( x_1 = 1.274 \)
• \( x_2 = -4.474 \)

These values can be verified by substituting them back into the original equation, ensuring that they both satisfy \( x^2 + 3.2x - 5.7 = 0 \). The roots provide valuable information about the points where the quadratic graph intersects the x-axis.