Question

Solve the equation. $$x^{2}+3 x-10=0$$

Short answer

The roots of the equation \(x^{2}+3 x-10=0\) are \(x = 2\) and \(x = -5\).

Step-by-step solution

Identify the coefficients

The quadratic equation has three coefficients: \(a\), \(b\), and \(c\). In the given equation, they are as follows: \(a = 1\), \(b = 3\), and \(c = -10\).

Compute the discriminant

The discriminant is the value under the square root in the quadratic formula. It is computed as \(b^2 - 4ac\). Hence, the discriminant = \(3^2 - 4*1*(-10) = 9 + 40 = 49\). This will indicate the nature of the roots. Positive discriminant means there are two real and distinct roots.

Apply the quadratic formula

The roots of the quadratic equation can be found by substituting the values of \(a\), \(b\), and the discriminant into the quadratic formula: \(x = [-b \pm sqrt(b^2 - 4ac)] /2a = [-3 \pm sqrt(49)]/2 = 2, -5\).

Key concepts

Quadratic Formula

One of the fundamental techniques for solving quadratic equations is the quadratic formula. A quadratic equation typically takes the form \( ax^2 + bx + c = 0 \) where \( a \) , \( b \) , and \( c \) are constants, and \( a \) is non-zero. The quadratic formula is an elegant solution that encompasses all possible types of solutions a quadratic equation can have. It states that the solutions for \( x \) are given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, the \( \pm \) symbol indicates that there are usually two solutions, one for \( + \) and another for \( - \) . Applying this formula allows one to calculate the roots without factoring or completing the square, thus making it a versatile tool, especially when equations are not easily factorable.

Discriminant

The discriminant is a vital component within the quadratic formula that gives valuable insight into the nature of the roots without actually calculating them. It is the part under the square root sign in the quadratic formula, determined by the formula \( D = b^2 - 4ac \) .

The value of the discriminant can tell us three things about the roots:

• If \( D > 0 \) , the equation has two real and distinct roots.
• If \( D = 0 \) , the equation has exactly one real root (or two real, identical roots).
• If \( D < 0 \) , the equation has two complex roots.

Understanding the discriminant is a powerful way to gauge the outcome of a quadratic equation quickly, and is particularly helpful when solving equations with real-world applications, where the type of roots can indicate certain possibilities or constraints in a given scenario.

Quadratic Equation Roots

The 'roots' of a quadratic equation are the values of \( x \) that satisfy the equation. In other words, they are the solutions that make the equation \( ax^2 + bx + c = 0 \) true. When using the quadratic formula to solve a quadratic equation, we find the roots by substituting our coefficients \( a \) , \( b \) , and \( c \) into the formula. As mentioned previously, the nature of these roots is indicated by the discriminant.

It's essential to grasp the significance of these roots in various contexts. For example, in physics, the roots could represent the points in time when a projectile is at a specific height. In economics, they could denote the break-even points for profits and losses. Different disciplines utilize the concept of quadratic roots in diverse ways, making it a critical concept to master for students across fields.