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Magazine Chapter 11: Problem 96
Use the quadratic formula to solve each quadratic equation. $$ 7 x^{2}+\sqrt{7} x-2=0 $$
Short Answer
Expert verified
The solutions are \(x = \frac{\sqrt{7}}{7}\) and \(x = -\frac{2\sqrt{7}}{7}\).
Step by step solution
01
Identify coefficients
The quadratic equation is given in the form \(ax^2 + bx + c = 0\). In the equation \(7x^2 + \sqrt{7}x - 2 = 0\), the coefficients are: \(a = 7\), \(b = \sqrt{7}\), and \(c = -2\).
02
Write the quadratic formula
The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula is used to find the roots of a quadratic equation.
03
Calculate the discriminant
The discriminant \(\Delta\) of a quadratic equation is \(b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\): \[\Delta = (\sqrt{7})^2 - 4 \times 7 \times (-2) = 7 + 56 = 63\]
04
Substitute values into the quadratic formula
Now we substitute \(a = 7\), \(b = \sqrt{7}\), and \(\Delta = 63\) into the quadratic formula:\[x = \frac{-\sqrt{7} \pm \sqrt{63}}{2 \times 7}\]
05
Simplify the roots
Simplify the expression by calculating \(\sqrt{63}\), which is \(\sqrt{9 \times 7} = 3\sqrt{7}\). Substitute this back:\[x = \frac{-\sqrt{7} \pm 3\sqrt{7}}{14}\]This gives us two potential solutions:\[x_1 = \frac{-\sqrt{7} +3\sqrt{7}}{14} = \frac{2\sqrt{7}}{14} = \frac{\sqrt{7}}{7}\]\[x_2 = \frac{-\sqrt{7} - 3\sqrt{7}}{14} = \frac{-4\sqrt{7}}{14} = -\frac{2\sqrt{7}}{7}\]
06
Final solutions
Our solutions are \(x_1 = \frac{\sqrt{7}}{7}\) and \(x_2 = -\frac{2\sqrt{7}}{7}\). These are the roots of the quadratic equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equation
A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\). This equation is called "quadratic" because it contains a squared term, specifically \(x^2\). Here, \(a\), \(b\), and \(c\) are known as coefficients and need to be real numbers. Suppose \(a\) is not zero, as this would make the equation linear instead of quadratic.
Quadratic equations often appear in algebra and describe parabolas when graphed on a coordinate plane.
The solution or "roots" of the quadratic equation can be found using various methods, one of which is the quadratic formula.
Quadratic equations often appear in algebra and describe parabolas when graphed on a coordinate plane.
- The coefficient \(a\) determines the direction and width of the parabola. If \(a\) is positive, the parabola opens upwards. If negative, it opens downwards.
- The coefficient \(b\) affects the position of the axis of symmetry of the parabola.
- Finally, \(c\) represents the y-intercept, where the parabola crosses the y-axis.
The solution or "roots" of the quadratic equation can be found using various methods, one of which is the quadratic formula.
Discriminant
The discriminant is a key player in understanding the nature of the roots of a quadratic equation. It is given by the expression \(b^2 - 4ac\) from the quadratic formula. Its value can provide information about the solutions of the equation without having to solve it entirely.
Here's how the discriminant helps determine the types of roots:
For the equation \(7x^2 + \sqrt{7}x - 2 = 0\), the calculated discriminant is 63, indicating two distinct real roots.
Here's how the discriminant helps determine the types of roots:
- If the discriminant is positive (\(b^2 - 4ac > 0\)), then the quadratic equation has two distinct real roots.
- If the discriminant is zero (\(b^2 - 4ac = 0\)), there is exactly one real root, sometimes referred to as a "double root," because the vertex of the parabola touches the x-axis.
- If the discriminant is negative (\(b^2 - 4ac < 0\)), there are no real roots. Instead, the roots are complex numbers.
For the equation \(7x^2 + \sqrt{7}x - 2 = 0\), the calculated discriminant is 63, indicating two distinct real roots.
Roots of an Equation
The roots of a quadratic equation are the solutions that make the equation equal to zero. These values of \(x\) are where the graph of the quadratic function intersects the x-axis. The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) helps us find these roots.
There are usually two roots, which may be either real numbers or complex numbers鈥攄epending on the discriminant. Let's explore this further:
In the example \(7x^2 + \sqrt{7}x - 2 = 0\), the discriminant is positive, hence the roots are \(x_1 = \frac{\sqrt{7}}{7}\) and \(x_2 = -\frac{2\sqrt{7}}{7}\). These are real and distinct solutions.
There are usually two roots, which may be either real numbers or complex numbers鈥攄epending on the discriminant. Let's explore this further:
- Two distinct real roots occur when the discriminant is positive.
- A single real root occurs when the discriminant is exactly zero.
- Two complex roots arise when the discriminant is negative.
In the example \(7x^2 + \sqrt{7}x - 2 = 0\), the discriminant is positive, hence the roots are \(x_1 = \frac{\sqrt{7}}{7}\) and \(x_2 = -\frac{2\sqrt{7}}{7}\). These are real and distinct solutions.
Quadratic Coefficients
Quadratic coefficients are the constants \(a\), \(b\), and \(c\) in a standard quadratic equation \(ax^2 + bx + c = 0\). Each coefficient has its role:
In our example \(7x^2 + \sqrt{7}x - 2 = 0\), the coefficients are \(a = 7\), \(b = \sqrt{7}\), and \(c = -2\). These coefficients are essential for substituting into the quadratic formula to find the roots and analyzing the parabola's behavior.
- \(a\) is the leading coefficient and cannot be zero. It determines the parabola's direction (upwards or downwards) and its width.
- \(b\) is the linear coefficient. It influences the parabola's symmetry and direction along the x-axis.
- \(c\) is the constant term, determining the parabola's intersection with the y-axis.
In our example \(7x^2 + \sqrt{7}x - 2 = 0\), the coefficients are \(a = 7\), \(b = \sqrt{7}\), and \(c = -2\). These coefficients are essential for substituting into the quadratic formula to find the roots and analyzing the parabola's behavior.
