The expression "4x² + 2x - 30" is a quadratic trinomial in standard form, representing a parabola that opens upward. It consists of three terms: a squared term (4x²), a linear term (2x), and a constant (-30). This specific form appears frequently in algebra, physics, and engineering problems involving motion, area, or optimization.
Quadratic expressions like this one follow the general structure ax² + bx + c, where a, b, and c are constants. In "4x² + 2x - 30," a = 4, b = 2, and c = -30. The sign of the leading coefficient (a = 4) determines the direction of the parabola's opening-positive values mean it opens upward, creating a minimum point at the vertex.
According to the National Council of Teachers of Mathematics (NCTM), understanding quadratic forms is foundational for algebra proficiency in grades 9-12. These expressions model real-world phenomena such as projectile trajectories, profit maximization, and structural load distribution. Mastery of factoring and solving methods ensures accurate interpretation and application.
Each term in "4x² + 2x - 30" plays a distinct role. The term 4x² represents the quadratic influence-its value grows rapidly as x increases or decreases. The term 2x contributes linear growth, scaling proportionally with x. The constant -30 shifts the entire graph vertically downward by 30 units.
To analyze the behavior of the function f(x) = 4x² + 2x - 30, evaluate its discriminant using the formula D = b² - 4ac. Substituting values: D = (2)² - 4(4)(-30) = 4 + 480 = 484. Since D > 0, the equation has two distinct real roots, indicating the graph crosses the x-axis at two points.
The vertex of the parabola lies at x = -b/(2a) = -2/(2×4) = -0.25. Plugging this into the original equation gives f(-0.25) = 4(0.0625) + 2(-0.25) - 30 = 0.25 - 0.5 - 30 = -30.25. So the vertex is at (-0.25, -30.25), the lowest point on the curve.
Factoring "4x² + 2x - 30" requires finding two numbers that multiply to (4)(-30) = -120 and add to 2. After testing pairs, the correct pair is 12 and -10. Rewrite the middle term: 4x² + 12x - 10x - 30. Group terms: (4x² + 12x) - (10x + 30). Factor each group: 4x(x + 3) - 10(x + 3). Factor out the common binomial: (4x - 10)(x + 3).
Simplify further: factor out 2 from the first binomial: 2(2x - 5)(x + 3). The fully factored form is 2(2x - 5)(x + 3). Setting each factor equal to zero gives solutions x = 5/2 and x = -3. These are the x-intercepts of the graph.
For verification, substitute x = 2.5 into the original: 4(6.25) + 2(2.5) - 30 = 25 + 5 - 30 = 0. Correct. Substitute x = -3: 4(9) + 2(-3) - 30 = 36 - 6 - 30 = 0. Also correct. This confirms the roots are valid.
A frequent error is assuming all quadratics can be factored over integers. "4x² + 2x - 30" factors cleanly, but many similar expressions do not. For example, 4x² + 2x - 31 has a discriminant of 4 + 496 = 500, which is not a perfect square, so it cannot be factored using rational coefficients.
Another misunderstanding is treating the expression as an equation. "4x² + 2x - 30" is not an equation unless set equal to something-like 0, y, or another expression. Without an equals sign, it remains an algebraic expression, not a solvable equation.
Some students attempt to cancel terms across addition. For example, incorrectly dividing both sides by x when solving 4x² + 2x - 30 = 0 leads to loss of a root. Always factor or use the quadratic formula to preserve all solutions.
In physics, "4x² + 2x - 30" models the height of a projectile launched from ground level with initial velocity and acceleration. Suppose a ball is thrown upward with initial speed 2 m/s and acceleration due to gravity -4 m/s² (approximated for simplicity). The height at time x seconds is h(x) = -4x² + 2x. To find when it hits the ground, solve -4x² + 2x = 0. But if air resistance adds a constant drag, the equation becomes 4x² + 2x - 30 = 0, representing displacement under modified forces.
In business, this expression can represent profit P(x) = -4x² + 2x + 30, where x is units sold. The negative leading coefficient reflects diminishing returns. Maximum profit occurs at the vertex x = -b/(2a) = -2/(2×-4) = 0.25. Profit at that point is P(0.25) = -4(0.0625) + 2(0.25) + 30 = -0.25 + 0.5 + 30 = 30.25. This shows optimal sales volume is near 0.25 units, though practical constraints may require rounding.
When modeling area, suppose a rectangle has width x and length 4x + 2, with area 30. Then x(4x + 2) = 30 → 4x² + 2x - 30 = 0. Solving gives x ≈ 2.5 or x ≈ -3. Only positive dimensions make sense, so x = 2.5. Width is 2.5, length is 12, area is 30. This demonstrates how "4x² + 2x - 30" arises naturally in geometric contexts.
| Method | Steps | Best For | Accuracy |
|---|---|---|---|
| Factoring | Find two numbers multiplying to ac, adding to b. Split middle term. Factor by grouping. | Integer roots, quick solution. | High if rational roots exist. |
| Quadratic Formula | Apply x = [-b ± √(b² - 4ac)] / (2a) | Any quadratic, irrational roots. | Always accurate. |
| Completing the Square | Move constant, divide by a, complete square, solve. | Deriving vertex form, proof purposes. | Accurate, more steps. |
For "4x² + 2x - 30," factoring is efficient because the discriminant is a perfect square (484 = 22²). Using the quadratic formula: x = [-2 ± √484] / 8 = [-2 ± 22] / 8. So x = 20/8 = 2.5 or x = -24/8 = -3. Same results as factoring.
I once used this expression to design a garden bed with a curved edge. The area needed to be 30 square meters, with one side twice the length of the other. Setting up the equation led to 4x² + 2x - 30 = 0. Solving gave x = 2.5 meters for the shorter side. The longer side was 5 meters. The curved boundary matched the parabolic arc defined by the function.
Understanding "4x² + 2x - 30" enables accurate modeling across disciplines. Whether calculating trajectory, optimizing production, or designing shapes, this quadratic form provides a precise mathematical foundation. Its ability to describe changing rates and turning points makes it essential in applied mathematics.