The roots of an equation are the values of the variable that make it true; for a polynomial equation set equal to 0, they are the solutions. The roots of x2 + 3x − 10 = 0 are 2 and −5, because each makes the left side 0. Real roots appear as the x-intercepts of the graph.
Factor, set each factor to 0, and check. Each root is a number that makes the whole equation true.
Formal definition and notation
A root of a polynomialp(x) is a numberr, real or complex, with p(r) = 0; equivalently, a solution of the equation p(x) = 0. By the factor theorem, r is a root exactly when x − r is a factor of p(x). A polynomial of degreen ≥ 1 has exactly n roots in the complex numbers when each is counted by its multiplicity.
A real root r is a number. The x-intercept is the point (r, 0) on the graph. Complex roots have no x-intercept.
Quick check: 3 questions
Question 1 of 3
What are the roots of (3x + 6)(x − 1) = 0?
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Answer: B.3x + 6 = 0 gives x = −2, and x − 1 = 0 gives 1. −6 forgot to divide by 3, and the last two have the signs reversed.
Question 2 of 3
Which equation has a double root at x = 4?
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Answer: D.x2 − 8x + 16 = (x − 4)2, so 4 is a root twice. x2 − 16 has roots ± 4, x2 + 8x + 16 has a double root at −4, and x2 − 4x has roots 0 and 4.
Question 3 of 3
The roots of x2 + bx + 12 = 0 are 2 and 6. What is b?
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Answer: A. The equation is (x − 2)(x − 6) = x2 − 8x + 12, so b = −8. The sum of the roots is −b = 8. 8 forgot the sign, 12 is the product, and −4 is 2 − 6.
Roots: frequently asked questions
What is the difference between roots and solutions?
For an equation set equal to 0, they are the same numbers. Root is the word used most for polynomial equations, as in “the roots of x2 − 9 = 0 are 3 and −3“. Solution is the general word for any value that makes any equation or inequality true, polynomial or not.
Can a quadratic equation have no real roots?
Yes. When the discriminant b2 − 4ac is negative, there are no real roots, and the parabola never meets the x-axis. For x2 + 2x + 5 = 0, b2 − 4ac = −16. The equation still has two complex roots, −1 ± 2i, which come in a conjugate pair.
What is a double root?
A double root is a root that comes from a squared factor, such as 5 in (x − 5)2 = 0. It counts as two roots that happen to be equal, which is why the discriminant is 0. On the graph, the parabola touches the x-axis at that point and turns back instead of crossing.
SolutionA solution is a value that makes an equation or inequality true. To test a value, substitute it for the variable and check that the statement holds: x = 4 solves x + 2 = 6 because 4 + 2 = 6. A solution can also be an ordered pair, a point on a graph that satisfies the relationship.Algebra · Grade 6, Grade 7, Grade 8PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7Complex NumberA complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, with i2 = −1. The real part is a and the imaginary part is b. Examples: 3 + 2i and −4i. Complex numbers let you take square roots of negatives, such as √−9 = 3i.Number System · Algebra 2, PrecalculusQuadratic EquationA quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2ParabolaA parabola is the U-shaped curve that is the graph of a quadratic function such as y = x2. It is symmetric about a line called the axis of symmetry, and its turning point is the vertex. Geometrically, it is every point equally far from a fixed point (the focus) and a fixed line (the directrix).Functions · Algebra 1, Algebra 2, GeometryX-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6, Grade 7SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1Cube RootThe cube root of a number is the value that, used as a factor three times, gives that number. It is written 3√x. Because 4 × 4 × 4 = 64, 3√64 = 4. Unlike square roots, cube roots of negative numbers are real numbers: 3√−27 = −3.Number System · Grade 8, Algebra 1, Algebra 2RadicalA radical is an expression that uses the root symbol √, such as √25 or 3√x. The number under the symbol is the radicand, and the small number in the notch is the index, which tells which root to take. With no index, it is a square root: √25 = 5.Number System · Grade 8, Algebra 1, Algebra 2Zero of a FunctionA zero of a function is an input that makes the output 0: a number a with f(a) = 0. For f(x) = x − 5, the zero is 5. On a graph, the real zeros are the x-values where the curve meets the x-axis. You find zeros by solving f(x) = 0.Functions · Algebra 1, Algebra 2, PrecalculusSquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3X-InterceptAn x-intercept is a point where a graph crosses or touches the x-axis. Its y-coordinate is always 0, so it is written (a, 0). To find it, set y = 0 and solve for x. A non-horizontal line has one x-intercept, while a parabola can have two, one or none.Algebra · Grade 8, Algebra 1, Algebra 2FactoringFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x2 + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Algebra · Grade 6, Grade 7, Grade 8Quadratic FormulaThe quadratic formula gives the solutions of any quadratic equation ax2 + bx + c = 0 with a ≠ 0: x = −b ± √b2 − 4ac2a. Substitute a, b and c, then compute once with plus and once with minus. It works even when the quadratic cannot be factored.Algebra · Algebra 1, Algebra 2DiscriminantThe discriminant of a quadratic equation ax2 + bx + c = 0 is the number b2 − 4ac. Its sign tells how many real solutions the equation has without solving it: positive means two, zero means one (a repeated root), and negative means none, because the parabola never touches the x-axis.Algebra · Algebra 1, Algebra 2