Question

Solve: \(5 x^{2}+2=5-14 x\)

Short answer

The solutions are \(x = 0.2\) and \(x = -3\).

Step-by-step solution

Move all terms to one side

Subtract 5 and add 14x to both sides of the equation to move all terms to one side: \[ 5x^{2} + 2 - 5 + 14x = 0 \]Simplify the equation: \[ 5x^{2} + 14x - 3 = 0 \]

Identify coefficients for the quadratic equation

The quadratic equation is of the form \(ax^2 + bx + c = 0\). Identify the coefficients:\(a = 5\), \(b = 14\), \(c = -3\)

Apply the quadratic formula

Use the quadratic formula to solve for \(x\): \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substitute the values of \(a\), \(b\), and \(c\): \[x = \frac{-14 \pm \sqrt{14^2 - 4(5)(-3)}}{2(5)}\]

Simplify under the square root

Simplify the terms under the square root: \[x = \frac{-14 \pm \sqrt{196 + 60}}{10}\]Simplify further: \[x = \frac{-14 \pm \sqrt{256}}{10}\]

Simplify the square root

Take the square root of 256: \[x = \frac{-14 \pm 16}{10}\]

Solve for both values of x

Calculate the two possible solutions: \[x = \frac{-14 + 16}{10} = \frac{2}{10} = 0.2\]And \[x = \frac{-14 - 16}{10} = \frac{-30}{10} = -3\]

Key concepts

quadratic formula

The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation has the form \(ax^2 + bx + c = 0\). The quadratic formula allows us to find the roots (solutions) of this equation. The formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here, \(a\), \(b\), and \(c\) are coefficients from the equation. The \(\pm\) symbol indicates two possible solutions.
For example, in the equation \(5x^2 + 14x - 3 = 0\), the coefficients are \(a = 5\), \(b = 14\), and \(c = -3\). Substituting these values into the quadratic formula gives us the potential solutions for \(x\).

simplifying equations

Simplifying equations is an important step in solving them. It involves combining like terms and reducing the equation to its simplest form.
For the equation \(5x^2 + 2 = 5 - 14x\):

• First, move all terms to one side by subtracting 5 and adding 14x on both sides:

\[5x^2 + 2 - 5 + 14x = 0\]
Combine the constants and like terms:

• The \(2 - 5\) becomes \(-3\), and the equation simplifies to:

\[5x^2 + 14x - 3 = 0\]
This equation is now in the standard quadratic form. Simplifying equations like this makes it easier to apply methods such as the quadratic formula for finding solutions.

algebraic manipulation

Algebraic manipulation involves rearranging and simplifying expressions and equations using algebraic rules. In solving quadratic equations, several algebraic steps might be required.
Feasibly, moving terms and changing signs might be the initial steps.
For example, in

• \(5x^2 + 2 = 5 - 14x\):

• We moved \(-5\) and \(14x\) to the left to get: \[5x^2 + 14x - 3 = 0\]

Following that, you'd need to identify coefficients:

• \(a = 5\), \(b = 14\), \(c = -3\)

The next steps involve inserting these into the quadratic formula and simplifying under the square root. Algebraic manipulation ensures each calculation step follows logically and correctly.

roots of quadratic equations

The roots of quadratic equations are the solutions for \(x\) that satisfy \(ax^2 + bx + c = 0\). These roots can be found using different methods, including factoring, completing the square, or the quadratic formula.
Using the quadratic formula to find roots involves several steps:

• First, substituting the coefficients into the formula.
• Simplifying under the square root.
• Calculating both potential values for \(x\).

For instance, in the equation \(5x^2 + 14x - 3 = 0\):

• We find the roots by solving:
• \[x = \frac{-14 \pm \sqrt{14^2 - 4(5)(-3)}}{2(5)}\]

After simplifying this expression, the results are:

• \(x = 0.2\)
• \(x = -3\)

These roots indicate where the quadratic equation intersects the \(x\)-axis on a graph.