Solve the given quadratic equations using the quadratic...

Solve the given quadratic equations using the quadratic formula. If there are no real roots, state this as the answer. Exercises $3-6$ are the same as Exercises $13-16$ of Section 7.2.
$$2 d(d-2)=-7$$

Key Concepts

Discriminant

The discriminant, given by the expression b² - 4ac in the quadratic formula, determines the nature of the roots of a quadratic equation. A positive discriminant indicates two distinct real roots, zero indicates one real repeated root, and a negative discriminant indicates the absence of real roots, instead leading to a pair of complex roots.

Quadratic Formula

The quadratic formula, x = (-b ± ?(b² - 4ac)) / (2a), is a universal method for solving any quadratic equation provided in standard form. It directly provides the solutions (or roots) of the quadratic equation by substituting the coefficients a, b, and c into the formula.

Standard Form of a Quadratic Equation

Expressing an equation in the standard form ax² + bx + c = 0 is a crucial step in solving quadratic equations. It involves rearranging the equation so that all terms are on one side of the equality sign, which then prepares the equation for application of solution methods like the quadratic formula.

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, generally written in the standard form ax² + bx + c = 0, where a, b, and c are constants with a ? 0. Recognizing an equation as quadratic allows one to apply specific solution strategies such as factoring, completing the square, or using the quadratic formula.

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