Question

Use the quadratic formula to solve the equation. $$-7 x^{2}-2.5 x+3=0$$

Short answer

The roots of the quadratic equation \( -7x^{2}-2.5x+3=0 \) are \( x_{1} = -0.145 \) and \( x_{2} = 2.855 \).

Step-by-step solution

Identify Coefficients

The coefficients are: \[\begin{align*} a & = -7, \ b & = -2.5, \ c & = 3. \end{align*}\]

Substitute Coefficients into the Quadratic Formula

Substitute \(a = -7\), \(b = -2.5,\) and \(c = 3\) into the quadratic formula \(x = [-b ± sqrt(b^{2}-4ac)] / (2a)\), to get: \[x = [2.5 ± sqrt((-2.5)^{2}-4*(-7)*3)] / (2*(-7)).\]

Simplify the Expression Under the Square Root

After simplification, the formula becomes: \[x = [2.5 ± sqrt(6.25+84)] / -14.\]

Further Simplify to Get the Result

After further simplification, the formula becomes: \[x = [2.5 ± sqrt(90.25)] / -14.\]

Calculating the Root Values

Calculate the value of \(x\) giving: \[\begin{align*} x_{1} & = (2.5 + sqrt(90.25)) / -14, \ x_{2} & = (2.5 - sqrt(90.25)) / -14. \end{align*}\]

Final Simplification to Get the Root Values

After simplifying, the roots of the given quadratic equation are: \[\begin{align*} x_{1} & = -0.145, \ x_{2} & = 2.855. \end{align*}\]

Key concepts

Solving Quadratic Equations

Solving quadratic equations is an essential skill in algebra. A quadratic equation is generally written in the form of ( ax^2 + bx + c = 0 ), where a, b, and c are known values, with a not equal to zero. To solve such equations, one can factorize the quadratic expression, complete the square, or as a more universally applicable method, use the quadratic formula, which is ( x = [-b ± sqrt(b^2-4ac)] / (2a) ). By substiting the coefficients a, b, and c from the original equation into the quadratic formula, the values of x can be calculated, which are the solutions to the equation.

When using the quadratic formula, it is important to evaluate the discriminant, which is the part under the square root, ( b^2-4ac ). The discriminant can indicate the nature of the roots (real and distinct, real and equal, or complex) and inform us about the characteristics of the parabola that represents the equation graphically.

Coefficients in Quadratic Equations

The coefficients in a quadratic equation ( ax^2 + bx + c = 0 ) play a pivotal role in determining the shape and position of the corresponding parabola on a graph. a is the leading coefficient and determines whether the parabola opens upwards (if a is positive) or downwards (if a is negative). b affects the position of the parabola along the x-axis, and c is the constant term that indicates the y-intercept of the parabola—the point where it crosses the y-axis.

Identifying coefficients is crucial as they are directly substituted into the quadratic formula when solving for the roots of the equation. In the example ( -7x^2 - 2.5x + 3 = 0 ), the coefficients are a = -7, b = -2.5, and c = 3. These values dictate the curvature and position of the parabola and ultimately lead to the precise calculation of the equation's solutions.

Simplifying Mathematical Expressions

The process of simplifying mathematical expressions is fundamental to solving equations efficiently. Simplification can include combining like terms, reducing fractions, and rationalizing denominators. When working with quadratic equations and the quadratic formula, it's important to simplify the expression under the square root (the discriminant) and the entire formula to identify the roots clearly.

In our example, after substiting the coefficients into the quadratic formula, the next critical step is to simplify the expression under the square root by performing the operations within: ( sqrt(b^2 - 4ac) ). Then, simplify the entire expression by combining like terms and simplifying any fractions. Clear simplification ensures the final step—root calculation—is as straightforward as possible, helping to minimize errors and misunderstandings.

Root Calculation of Quadratic Equations

Root calculation in quadratic equations involves finding the values of x that satisfy the equation ( ax^2 + bx + c = 0 ). Once the quadratic formula has been simplified, the final step is to calculate the root values. This involves adding and subtracting the square root value obtained from the discriminant to/from -b, and then dividing each result by ( 2a ). This yields two potential values for x, which are the solutions or the 'roots' of the original equation.

In our example, with the equation ( -7x^2 - 2.5x + 3 = 0 ), after simplification, we are left with the roots ( x_1 and x_2 ). To obtain these values, we take the square root part of the simplified quadratic formula, perform the addition and subtraction, and then carry out the division, yielding two values for x. These values can be interpreted as the points where the parabola represented by the quadratic equation intersects the x-axis.